Algebra 1 · Class 11 · Grades 7–8

Algebra 1 Class 11: Polynomials — Adding, Subtracting, Multiplying — 10 SOMATH Posters, Theory & 70 Practice Questions

The whole Algebra 1 polynomial-operations unit in one class. Ten SOMATH posters walk through what a polynomial is, how to identify terms, coefficients, and constants, how to classify polynomials, how to find the degree, combining like terms, adding, subtracting, multiplying monomials, distributive multiplication, and FOIL for two binomials. Under each poster you’ll find full theory notes and 5 progressively harder practice questions with click-to-reveal step-by-step solutions — plus 20 final mixed-review questions, for 70 questions total.

· By SOMATH · Class 11 of the Algebra 1 (Algebra 1) rolling syllabus · (646) 668-6151 · 226 W 79th St, New York, NY 10024

Class 11 of the SOMATH Algebra 1 course

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Short answer. To add polynomials, combine like terms (same variable, same exponent). To subtract, distribute the minus sign to every term of the second polynomial and then combine like terms. To multiply monomials, multiply the coefficients and add exponents of the same base (xm · xn = xm+n). To multiply a monomial by a polynomial, use the distributive property. To multiply two binomials, use FOIL — First, Outer, Inner, Last.

What’s in this class

  1. Introduction to Polynomials
  2. Identifying Terms, Coefficients, and Constants
  3. Classifying Polynomials (Monomial, Binomial, Trinomial)
  4. Degree of a Polynomial
  5. Combining Like Terms
  6. Adding Polynomials
  7. Subtracting Polynomials
  8. Multiplying Monomials
  9. Multiplying a Monomial by a Polynomial (Distributive)
  10. Multiplying Two Binomials — The FOIL Method
  11. 20 Final Questions: Mixed Review
How to use this page. Study each poster first, then read the theory notes below it, then work the 5 practice questions. Each set builds from a short warm-up to an intermediate application. Finish with the 20-question mixed review covering all ten topics. Click Answer only after you’ve tried a problem. Bring questions to your SOMATH Algebra 1 teacher — or book a free 30-minute evaluation at 226 W 79th St.

1. Introduction to Polynomials

SOMATH Algebra 1 Class 11 poster — Introduction to Polynomials, showing variables, coefficients, exponents, and the example 3x² + 2x − 5

Definition

A polynomial is an algebraic expression made of variables, coefficients, and constants combined only by addition, subtraction, and multiplication — and every variable exponent is a whole number (0, 1, 2, 3, …).

Example

In 3x2 + 2x − 5:

  • Variable: x.
  • Coefficients: 3 (of x2) and 2 (of x).
  • Constant: −5.
  • Exponents: 2 and 1 — both whole numbers, so this is a polynomial.

What is NOT a polynomial

Expressions with variables in a denominator (1/x), under a radical (√x), or with negative or fractional exponents (x−2, x1/2) are not polynomials.

Why it matters

Every algebra topic that follows — factoring, quadratics, rational expressions, functions — starts with fluent polynomial operations.

5 Practice Questions — Introduction to Polynomials

Q1. Is 5x³ − 2x² + x − 8 a polynomial? Identify its variable, coefficients, and constant.
Answer

Yes. Variable: x; coefficients: 5, −2, and 1; constant: −8.

  1. The exponents 3, 2, and 1 are nonnegative integers, so this is a polynomial.
  2. The term x has coefficient 1. Keep the minus signs with −2 and −8.
Q2. Which expression is a polynomial: A = 5x² − 3x + 1 or B = 5x² − 3/x + 1? Explain.
Answer

A is a polynomial; B is not.

  1. A contains only whole-number powers of x.
  2. In B, x is in a denominator. The term 3/x has a negative variable exponent, which is not allowed.
Q3. A student says √x + 4 and x² + 4 are both polynomials. Is the student correct?
Answer

No. Only x² + 4 is a polynomial.

  1. The square root of x has exponent 1/2, not a whole-number exponent.
  2. The expression x² + 4 has exponent 2 and a constant, so it is a polynomial.
Q4. In −3x²y + 4xy − 6, identify the variables, numerical coefficients, and constant. Explain why two variables are allowed.
Answer

Variables: x and y; coefficients: −3 and 4; constant: −6. It is a polynomial.

  1. A polynomial may have more than one variable.
  2. Here all variable exponents are nonnegative integers: x²y has exponents 2 and 1, while xy has exponents 1 and 1.
Q5. Simplify 2x² + 3x − x² + 4. Is the result a polynomial? State its degree.
Answer

x² + 3x + 4; yes, it is a polynomial of degree 2.

  1. Combine the like terms 2x² − x² = x².
  2. The highest remaining exponent is 2. The linear term and constant stay separate.

2. Identifying Terms, Coefficients, and Constants

SOMATH Class 11 poster — Identifying Terms, Coefficients, and Constants in 4x² − 7x + 9

Terms

A term is a piece of a polynomial separated by + or − signs. The sign in front belongs to the term.

Coefficient

The coefficient is the number multiplying the variable in a term. If a variable has no visible number, the coefficient is 1 (or −1 if the term is negative).

Constant

The constant is the term with no variable at all.

Worked example: 4x2 − 7x + 9

  • Terms: 4x2, −7x, 9.
  • Coefficients: 4 (of x2) and −7 (of x).
  • Constant: 9.

5 Practice Questions — Terms, Coefficients, and Constants

Q1. List the signed terms of 4x³ − 2x² + 5x − 9. Identify the constant.
Answer

Terms: 4x³, −2x², 5x, and −9. The constant is −9.

  1. Separate the expression into pieces joined by addition or subtraction.
  2. Keep each subtraction sign with the term that follows it. The constant has no variable.
Q2. For −6x³ + x − 7, give the coefficients of x³, x², and x, then the constant.
Answer

The coefficients are −6, 0, and 1; the constant is −7.

  1. There is no x² term, so its coefficient is 0.
  2. The unwritten coefficient of x is 1. The constant is the signed number −7.
Q3. In 3a²b − 5ab + 8, identify the numerical coefficients, constant, and number of terms.
Answer

Coefficients: 3 and −5; constant: 8; three terms.

  1. The variable terms are 3a²b and −5ab, so their numerical coefficients are 3 and −5.
  2. The standalone 8 is the constant and counts as a third term.
Q4. Simplify 5x² − 2x + 3 − x², then identify the coefficient of x², coefficient of x, and constant.
Answer

4x² − 2x + 3; coefficients 4 and −2; constant 3.

  1. Combine 5x² − x² = 4x².
  2. The terms −2x and 3 remain unchanged. Read the coefficients after simplifying.
Q5. Write 7 − x³ + 2x² − 4x in descending powers of x. Identify its leading coefficient and constant.
Answer

−x³ + 2x² − 4x + 7; leading coefficient −1; constant 7.

  1. Move the highest power first, keeping every term's sign.
  2. The leading term is −x³, whose coefficient is −1. The constant remains 7.

3. Classifying Polynomials by Number of Terms

SOMATH Class 11 poster — Classifying polynomials: monomial (6x²), binomial (x + 5), trinomial (2x² − 3x + 4)

The three named types

# of termsNameExample
1Monomial6x2
2Binomialx + 5
3Trinomial2x2 − 3x + 4
4+Polynomial (no special name)x3 + 2x2 − x + 7

Simplify first

Always combine like terms before classifying. 3x + 5x + 2 looks like 3 terms but simplifies to 8x + 2 — a binomial.

5 Practice Questions — Classifying Polynomials

Q1. Simplify and classify by number of terms: 7x² − 3x + 2x² + 8x.
Answer

9x² + 5x, a binomial.

  1. Combine the two quadratic terms and the two linear terms separately.
  2. Two nonzero terms remain; do not count the four terms in the original expression.
Q2. Simplify and classify: 5y³ + 2y² − 4y − 5y³ + 7y.
Answer

2y² + 3y, a binomial.

  1. The y³ terms cancel. The linear terms combine to −4y + 7y = 3y.
  2. The remaining two unlike terms cannot be combined.
Q3. Simplify and classify by number of terms: 3x² + 2x − x² + 5.
Answer

2x² + 2x + 5, a trinomial.

  1. Combine 3x² − x² = 2x².
  2. Three unlike, nonzero terms remain, so the simplified polynomial is a trinomial.
Q4. Simplify and classify 4x² + 3x − 4x² − 3x + 5.
Answer

5, a monomial.

  1. Both variable pairs cancel: 4x² − 4x² = 0 and 3x − 3x = 0.
  2. The nonzero constant 5 is one term, so it is a monomial.
Q5. A student calls 6y² − 2y + 4 − 3y² + 5y a five-term polynomial. Simplify it and correct the classification.
Answer

3y² + 3y + 4, a trinomial.

  1. Combine 6y² − 3y² = 3y² and −2y + 5y = 3y.
  2. Classification uses the simplified expression. The constant 4 gives a total of three terms.

4. Degree of a Polynomial

SOMATH Class 11 poster — Degree of a polynomial: monomial degree = sum of exponents (5x³y² → 5); polynomial degree = largest term degree (4x⁴ − 2x² + 7 → 4)

Degree of a monomial

Add the exponents of all variables in that term. 5x3y2 has degree 3 + 2 = 5.

Degree of a polynomial

The largest degree among all its terms. 4x4 − 2x2 + 7 has degree 4.

Special cases

  • A nonzero constant has degree 0 (because c = c · x0).
  • The zero polynomial (just 0) has an undefined degree — it’s a special case.

Why degree matters

Degree tells you how the polynomial behaves and how many roots it can have. Degree 1 = linear, 2 = quadratic, 3 = cubic, and so on.

5 Practice Questions — Degree of a Polynomial

Q1. Find the degree and leading coefficient of 8 − 3x⁴ + 5x² − x.
Answer

Degree 4; leading coefficient −3.

  1. In standard form it is −3x⁴ + 5x² − x + 8.
  2. The highest exponent is 4. Its coefficient is −3, not the first number shown in the original order.
Q2. Find the total degree of 4x³y + 2xy² − 5. Show each term's degree.
Answer

The term degrees are 4, 3, and 0; the total degree is 4.

  1. For x³y, add 3 + 1 = 4. For xy², add 1 + 2 = 3.
  2. The nonzero constant has degree 0. Take the largest term degree.
Q3. Find the degree of 7x² − 3x⁴ + 6x − 2, and identify the term that determines it.
Answer

Degree 4, determined by −3x⁴.

  1. Look for the largest exponent, even when the terms are not in standard order.
  2. The negative coefficient does not change the degree of the term.
Q4. Find the total degree of 4a³b² − 2ab⁴ + 7. Why is it not 10?
Answer

The total degree is 5.

  1. The first two term degrees are 3 + 2 = 5 and 1 + 4 = 5.
  2. Use the largest individual term degree, not the sum of the degrees of different terms.
Q5. Simplify 5x⁴ + 2x² − 3x − 5x⁴ + 7 before finding its degree.
Answer

2x² − 3x + 7; degree 2.

  1. The terms 5x⁴ and −5x⁴ cancel.
  2. The highest surviving power is x², so the degree drops to 2.

5. Combining Like Terms

SOMATH Class 11 poster — Combining like terms: 3x² + 5x² = 8x²; simplify 4x² + 3x − 2x² + 5x − 7 to 2x² + 8x − 7

What are like terms?

Like terms have the same variables raised to the same powers. Only the coefficients can differ. Examples of like terms: 3x2 and 5x2. 2x and 2x2 are not like terms — the exponents differ.

How to combine

Add (or subtract) the coefficients. Keep the variable and exponent unchanged: 3x2 + 5x2 = 8x2.

Worked example

Simplify 4x2 + 3x − 2x2 + 5x − 7:

  • Group: (4x2 − 2x2) + (3x + 5x) − 7.
  • Combine: 2x2 + 8x − 7.

Common pitfalls

  • Don’t add exponents when combining like terms — you only add coefficients.
  • The sign in front belongs to the term. −2x2 stays negative when you group it.

5 Practice Questions — Combining Like Terms

Q1. Simplify 8x − 3 + 5x − 7 − 4x.
Answer

9x − 10.

  1. The x-coefficient is 8 + 5 − 4 = 9.
  2. The constants total −3 − 7 = −10.
Q2. Simplify 6x² − 3x + 4 − 2x² + 5x.
Answer

4x² + 2x + 4.

  1. Combine the quadratic terms: 6x² − 2x² = 4x².
  2. Combine −3x + 5x = 2x. The constant 4 stays unchanged.
Q3. Simplify 4x + 3y − x + 2y − 6.
Answer

3x + 5y − 6.

  1. Combine the x-terms: 4x − x = 3x. Combine the y-terms: 3y + 2y = 5y.
  2. Do not combine x-terms with y-terms. The constant remains −6.
Q4. Simplify and write in standard form: 3x³ + 2x² − x + x³ − 5x² + 4x.
Answer

4x³ − 3x² + 3x.

  1. Combine equal powers separately: (3 + 1)x³, (2 − 5)x², and (−1 + 4)x.
  2. The coefficients become 4, −3, and 3.
Q5. Simplify 3x² + 2x − x² + 5, then evaluate at x = 2.
Answer

2x² + 2x + 5; at x = 2, the value is 17.

  1. Combine 3x² − x² = 2x².
  2. Substitute x = 2: 2(2²) + 2(2) + 5 = 8 + 4 + 5 = 17.

6. Adding Polynomials

SOMATH Class 11 poster — Adding polynomials: (3x² + 2x − 5) + (2x² − x + 7) = 5x² + x + 2, shown horizontally and with the vertical column method

The rule

To add polynomials, combine only like terms. Signs stay the same — you’re just removing parentheses and grouping.

Worked example (horizontal)

Add (3x2 + 2x − 5) + (2x2 − x + 7):

  1. Group like terms: (3x2 + 2x2) + (2x − x) + (−5 + 7).
  2. Add coefficients: 5x2 + x + 2.

Vertical method

Line up like terms in columns, then add each column:

    3x² + 2x − 5
  + 2x² −  x + 7
  ————————
    5x² +  x + 2

Tip

Line up terms of the same degree even when a degree is missing — leave a blank space so the columns stay clean.

5 Practice Questions — Adding Polynomials

Q1. Add and simplify: (4x² − 7x + 3) + (−2x² + 5x − 8).
Answer

2x² − 2x − 5.

  1. Combine like terms: (4 − 2)x² + (−7 + 5)x + (3 − 8).
  2. This gives 2x² − 2x − 5. Adding a polynomial does not change its existing signs.
Q2. Add (2x² + 3x − 4) + (x² − 5x + 7).
Answer

3x² − 2x + 3.

  1. Add matching coefficients: 2 + 1 = 3 and 3 − 5 = −2.
  2. Add constants: −4 + 7 = 3.
Q3. Add and simplify (4x² − 2x + 6) + (−4x² + 5x − 1).
Answer

3x + 5.

  1. The quadratic terms cancel: 4x² − 4x² = 0.
  2. The remaining terms give (−2 + 5)x + (6 − 1) = 3x + 5.
Q4. Add (2x³ + 4x − 3) + (x³ + 2x² − x + 5).
Answer

3x³ + 2x² + 3x + 2.

  1. Combine the cubic terms and the linear terms: 2x³ + x³ = 3x³ and 4x − x = 3x.
  2. The quadratic term 2x² has no partner. The constants total −3 + 5 = 2.
Q5. A triangle has side lengths 2x + 5, 3x − 1, and x + 4 centimeters, with x > 1. Write and simplify its perimeter, then find the perimeter when x = 3.
Answer

P(x) = 6x + 8 cm; at x = 3, the perimeter is 26 cm.

  1. Add the three sides: (2x + 5) + (3x − 1) + (x + 4) = 6x + 8.
  2. Evaluate 6(3) + 8 = 26. The side lengths are then 11, 8, and 7 cm.

7. Subtracting Polynomials

SOMATH Class 11 poster — Subtracting polynomials: (5x² + 3x − 4) − (2x² − x + 6) becomes 5x² + 3x − 4 − 2x² + x − 6, which simplifies to 3x² + 4x − 10

The rule

Subtracting = adding the opposite. Distribute the minus sign to every term in the second polynomial — flip every sign — then combine like terms.

Worked example

Compute (5x2 + 3x − 4) − (2x2 − x + 6):

  1. Distribute the minus: 5x2 + 3x − 4 − 2x2 + x − 6.
  2. Group like terms: (5x2 − 2x2) + (3x + x) + (−4 − 6).
  3. Combine: 3x2 + 4x − 10.

Common pitfall

Students flip only the first sign of the second polynomial. You must flip every sign inside those parentheses. Watch double negatives carefully: −(−x) = +x.

5 Practice Questions — Subtracting Polynomials

Q1. Subtract (5x² + 3x − 2) − (2x² + x + 4).
Answer

3x² + 2x − 6.

  1. Change all signs in the second polynomial: 5x² + 3x − 2 − 2x² − x − 4.
  2. Combine matching powers and constants: 3x² + 2x − 6.
Q2. Simplify (6x² − 2x + 5) − (3x² − 4x + 1).
Answer

3x² + 2x + 4.

  1. Subtracting −4x produces +4x: 6x² − 2x + 5 − 3x² + 4x − 1.
  2. Combine to get 3x² + 2x + 4.
Q3. Simplify (3x³ + 2x − 4) − (3x³ − x + 5).
Answer

3x − 9.

  1. Distribute the subtraction: 3x³ + 2x − 4 − 3x³ + x − 5.
  2. The cubic terms cancel. The linear terms give 3x, and the constants give −9.
Q4. Subtract and simplify (−2x² + 5x − 1) − (x² − 3x + 4).
Answer

−3x² + 8x − 5.

  1. Rewrite as −2x² + 5x − 1 − x² + 3x − 4.
  2. Combine carefully: −2 − 1 = −3, 5 + 3 = 8, and −1 − 4 = −5.
Q5. A school club earns 12x + 8 dollars from selling x items and spends 5x + 20 dollars. Write the profit expression, then find the profit when x = 4.
Answer

7x − 12; the profit at x = 4 is $16.

  1. Profit is earnings minus spending: (12x + 8) − (5x + 20) = 7x − 12.
  2. Substitute 4: 7(4) − 12 = 28 − 12 = 16.

8. Multiplying Monomials

SOMATH Class 11 poster — Multiplying monomials: (3x²)(4x³) = 12x⁵ using the rule xᵐ · xⁿ = x^(m+n); (−2a³b)(5a²b⁴) = −10a⁵b⁵

Two-step rule

  1. Multiply the coefficients (just numbers).
  2. Add the exponents of matching bases: xm · xn = xm+n.

Example 1

(3x2)(4x3) = (3 · 4)(x2+3) = 12x5.

Example 2 — two variables

(−2a3b)(5a2b4): Coefficients: (−2)(5) = −10. a-exponents: 3 + 2 = 5. b-exponents: 1 + 4 = 5. Answer: −10a5b5.

Remember

A variable with no visible exponent has exponent 1 (so b = b1). Only add exponents when the bases are the same.

5 Practice Questions — Multiplying Monomials

Q1. Multiply (−3x²)(5x³), then state the degree of the product.
Answer

−15x⁵; degree 5.

  1. Multiply coefficients: (−3)(5) = −15.
  2. Add exponents of the same base: x² · x³ = x⁵.
Q2. Multiply (−2a³b²)(−4ab³). Give the total degree of the product.
Answer

8a⁴b⁵; total degree 9.

  1. Two negative coefficients make a positive: (−2)(−4) = 8.
  2. Add exponents by variable: a³ · a = a⁴ and b² · b³ = b⁵. Total degree is 4 + 5 = 9.
Q3. Multiply (3x²y)(−2xy²).
Answer

−6x³y³.

  1. Multiply coefficients: 3(−2) = −6.
  2. Add exponents for each variable: x² · x = x³ and y · y² = y³.
Q4. Multiply the three factors (−2x)(3x²)(−x).
Answer

6x⁴.

  1. Multiply coefficients: (−2)(3)(−1) = 6.
  2. Add the x-exponents: 1 + 2 + 1 = 4.
Q5. A rectangle has length 4x² centimeters and width 3x centimeters, where x > 0. Write its area as a monomial and state the degree.
Answer

12x³ cm²; degree 3.

  1. Multiply length by width: (4x²)(3x).
  2. Multiply 4 · 3 = 12 and add exponents 2 + 1 = 3.

9. Multiplying a Monomial by a Polynomial (Distributive Property)

SOMATH Class 11 poster — Multiplying a monomial by a polynomial using distributive property: 3x(2x² − 5x + 4) = 6x³ − 15x² + 12x

The rule

Use the distributive property: a(b + c) = ab + ac. Multiply the monomial by every term inside the parentheses.

Example 1

3x(2x2 − 5x + 4):

  • 3x · 2x2 = 6x3
  • 3x · (−5x) = −15x2
  • 3x · 4 = 12x

Answer: 6x3 − 15x2 + 12x.

Example 2 — negative monomial

−2y2(3y3 + y − 6):

  • −2y2 · 3y3 = −6y5
  • −2y2 · y = −2y3
  • −2y2 · (−6) = 12y2

Answer: −6y5 − 2y3 + 12y2.

Warning about signs

A negative monomial flips the sign of every term. Apply the sign rules of multiplication carefully — missing a sign is the #1 error on this topic.

5 Practice Questions — Monomial × Polynomial

Q1. Expand 2x(3x − 4).
Answer

6x² − 8x.

  1. Multiply 2x by both terms.
  2. 2x · 3x = 6x² and 2x · (−4) = −8x.
Q2. Expand 3x(2x² − x + 4).
Answer

6x³ − 3x² + 12x.

  1. Distribute 3x to all three terms, not just the first.
  2. The products are 6x³, −3x², and 12x.
Q3. Expand −2x(3x² − 4x + 5).
Answer

−6x³ + 8x² − 10x.

  1. Multiply −2x by each term.
  2. The middle product is positive because (−2x)(−4x) = 8x². The other two products are negative.
Q4. Expand and then combine like terms: 2x(x + 3) + x².
Answer

3x² + 6x.

  1. Distribute first: 2x(x + 3) = 2x² + 6x.
  2. Add the extra x²: 2x² + x² = 3x².
Q5. A rectangle has width 2x meters and length x + 4 meters, where x > 0. Write its area as a polynomial, then find the area when x = 3.
Answer

2x² + 8x m²; when x = 3, the area is 42 m².

  1. Area is width times length: 2x(x + 4) = 2x² + 8x.
  2. Evaluate 2(3²) + 8(3) = 18 + 24 = 42.

10. Multiplying Two Binomials — The FOIL Method

SOMATH Class 11 poster — FOIL method for multiplying two binomials: (x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15

FOIL — First, Outer, Inner, Last

FOIL is a memory aid for the distributive property applied to two binomials. You get four products:

  • First — the first term of each binomial.
  • Outer — the two outer terms.
  • Inner — the two inner terms.
  • Last — the last term of each binomial.

Worked example: (x + 3)(x + 5)

  • First: x · x = x2
  • Outer: x · 5 = 5x
  • Inner: 3 · x = 3x
  • Last: 3 · 5 = 15

Sum: x2 + 5x + 3x + 15. Combine like terms: x2 + 8x + 15.

Why FOIL works

FOIL is just the distributive property in disguise. It only works for two binomials. For a binomial times a trinomial — or any bigger product — use the distributive property directly and multiply every term of the first by every term of the second.

Sign patterns worth memorizing

  • (x + a)(x + b) = x2 + (a+b)x + ab
  • (x + a)(x − a) = x2 − a2 — the difference of squares.
  • (x + a)2 = x2 + 2ax + a2 — the perfect square.

5 Practice Questions — FOIL / Multiplying Binomials

Q1. Expand (x + 3)(x + 7).
Answer

x² + 10x + 21.

  1. The four products are x², 7x, 3x, and 21.
  2. Combine the two middle terms: 7x + 3x = 10x.
Q2. Expand and simplify (x − 4)(x + 6).
Answer

x² + 2x − 24.

  1. Multiply every term: x² + 6x − 4x − 24.
  2. Combine 6x − 4x = 2x. Keep the negative constant −24.
Q3. Expand and simplify (2x + 3)(x + 4).
Answer

2x² + 11x + 12.

  1. The four products are 2x², 8x, 3x, and 12.
  2. The middle terms combine to 11x.
Q4. Expand (x − 5)². Explain why the answer is not x² + 25.
Answer

x² − 10x + 25.

  1. Write the square as (x − 5)(x − 5). The four products are x², −5x, −5x, and 25.
  2. The two middle terms add to −10x. Squaring a binomial does not mean squaring only its two terms.
Q5. A rectangular poster measures x + 2 centimeters by x + 5 centimeters, where x > 0. Write its area as a polynomial, then find the area when x = 3.
Answer

x² + 7x + 10 cm²; when x = 3, the area is 40 cm².

  1. Multiply the dimensions: (x + 2)(x + 5) = x² + 5x + 2x + 10 = x² + 7x + 10.
  2. Substitute x = 3: 9 + 21 + 10 = 40. Check with the dimensions 5 cm by 8 cm.

20 Final Questions: All 10 Polynomial Topics

Start with identifying polynomials, terms, and degree, then work toward combining operations and solving short real-world problems. These 20 new questions gradually build from foundations to intermediate challenges and review all ten posters. Try each one before opening its answer.

F1. Which expression is a polynomial: A = 3x² + 2x − 9 or B = 3/x + 2x − 9? Explain your choice.
Answer

Only A is a polynomial.

  1. A uses nonnegative integer powers of x: 2, 1, and 0.
  2. B contains x in a denominator: 3/x = 3x⁻¹. A negative variable exponent is not allowed in a polynomial.
F2. For 7x³ − 4x + 6, list the signed terms, identify the constant, and give the coefficient of x².
Answer

Terms: 7x³, −4x, and 6; constant: 6; coefficient of x²: 0.

  1. Keep each sign attached to its term.
  2. There is no x² term, so its coefficient is 0. The constant is the term without a variable.
F3. Classify each expression as a monomial, binomial, or trinomial: −8x²; 2x³ + 5; x² − 6x + 9.
Answer

Monomial; binomial; trinomial, respectively.

  1. Count nonzero terms separated by addition or subtraction.
  2. The expressions have one, two, and three terms. Exponents do not determine the number of terms.
F4. Write 5 − 2x + 4x³ − x² in standard form. State its degree and leading coefficient.
Answer

4x³ − x² − 2x + 5; degree 3; leading coefficient 4.

  1. Arrange terms in descending powers, keeping their signs.
  2. The leading term is 4x³. Its exponent gives the degree and its numerical factor gives the leading coefficient.
F5. Find the degree of 2a²b² − 5ab + 3. Explain why the answer is not 2.
Answer

Degree 4.

  1. Add the variable exponents within each term: a²b² has degree 2 + 2 = 4, and ab has degree 1 + 1 = 2.
  2. The constant has degree 0. The largest term degree is 4; looking at only one exponent would miss the second variable.
F6. Simplify 7x² − 4x + 2 − 3x² + 6x, then classify the result by number of terms.
Answer

4x² + 2x + 2, a trinomial.

  1. Combine square terms: 7x² − 3x² = 4x².
  2. Combine linear terms: −4x + 6x = 2x. Keep the constant 2, leaving three terms.
F7. Add (3x² + 2x − 5) + (2x² − 7x + 8).
Answer

5x² − 5x + 3.

  1. Add the x² coefficients: 3 + 2 = 5.
  2. Add the x coefficients: 2 − 7 = −5. Add the constants: −5 + 8 = 3.
F8. Subtract (8x² − 3x + 4) − (5x² + 2x − 6).
Answer

3x² − 5x + 10.

  1. Distribute the minus sign: 8x² − 3x + 4 − 5x² − 2x + 6.
  2. Combine like terms. In particular, subtracting −6 adds 6, so the constant becomes 10.
F9. Multiply (−4x²)(3x³). State the degree of your answer.
Answer

−12x⁵, degree 5.

  1. Multiply coefficients: −4 × 3 = −12.
  2. Add exponents for the same base: x² × x³ = x⁵. The result has degree 5.
F10. Expand −3x(2x² − x + 4).
Answer

−6x³ + 3x² − 12x.

  1. Multiply −3x by every term, not just the first.
  2. The products are −6x³, +3x², and −12x. A negative times a negative gives the positive middle term.
F11. Expand (x + 4)(x − 7), then state its degree and constant.
Answer

x² − 3x − 28; degree 2; constant −28.

  1. The four products are x², −7x, 4x, and −28.
  2. Combine −7x + 4x = −3x. The highest exponent is 2 and the constant is −28.
F12. Simplify 6x³ + 4x² − x − 6x³ − x² + 5x. Classify the result and explain how its degree changes.
Answer

3x² + 4x, a binomial of degree 2.

  1. The cubic terms cancel: 6x³ − 6x³ = 0.
  2. Combine 4x² − x² = 3x² and −x + 5x = 4x. Although cubic terms appeared in the original expression, the simplified polynomial has degree 2.
F13. Let P = 4x² − x + 7 and Q = 2x² + 3x − 5. Find P − Q, then evaluate it at x = 2.
Answer

P − Q = 2x² − 4x + 12; at x = 2, the value is 12.

  1. Subtract every term of Q: 4x² − x + 7 − 2x² − 3x + 5 = 2x² − 4x + 12.
  2. Substitute 2: 2(4) − 4(2) + 12 = 8 − 8 + 12 = 12.
F14. Multiply (−3a²b)(2ab²). Identify its coefficient, number of terms, and total degree.
Answer

−6a³b³; coefficient −6; one term; total degree 6.

  1. Multiply coefficients and add matching exponents: −3 × 2 = −6, a² × a = a³, and b × b² = b³.
  2. It is one monomial. Add the exponents within that term: 3 + 3 = 6.
F15. Simplify 3x(x − 2) + 2x² + 5x. State its degree and classify it by number of terms.
Answer

5x² − x, a binomial of degree 2.

  1. Distribute first: 3x(x − 2) = 3x² − 6x.
  2. Combine 3x² + 2x² = 5x² and −6x + 5x = −x. Two terms remain, with highest exponent 2.
F16. Expand (3x − 2)(x + 5), then evaluate your polynomial at x = 2.
Answer

3x² + 13x − 10; at x = 2, the value is 28.

  1. Multiply to get 3x² + 15x − 2x − 10, then combine the middle terms.
  2. Evaluate: 3(4) + 13(2) − 10 = 28. Check in the original form: (6 − 2)(2 + 5) = 4 × 7 = 28.
F17. A student claims (x + 6)² = x² + 36. Correct the expansion, explain the missing term, and evaluate the correct result at x = 2.
Answer

x² + 12x + 36; at x = 2, the value is 64.

  1. Write (x + 6)(x + 6). The four products are x², 6x, 6x, and 36.
  2. The two middle products add to 12x; they cannot be omitted.
  3. At x = 2: 4 + 24 + 36 = 64, matching (2 + 6)².
F18. A school club sells x craft kits for $3x each. Its total cost is 2x² + 4x + 6 dollars. Write a polynomial for profit, then find the profit or loss when x = 6.
Answer

Profit = x² − 4x − 6 dollars; at x = 6, the club makes a $6 profit.

  1. Revenue equals quantity times price: x(3x) = 3x².
  2. Profit = revenue − cost = 3x² − (2x² + 4x + 6) = x² − 4x − 6.
  3. At x = 6: 36 − 24 − 6 = 6. A positive result means a profit.
F19. A rectangular display measures 2x + 1 feet by x + 3 feet, where x > 0. Write simplified polynomials for its perimeter and area, then find both when x = 2.
Answer

Perimeter = 6x + 8 feet; area = 2x² + 7x + 3 square feet. At x = 2: perimeter 20 feet and area 25 square feet.

  1. Perimeter adds all four sides: 2(2x + 1) + 2(x + 3) = 6x + 8.
  2. Area multiplies the dimensions: (2x + 1)(x + 3) = 2x² + 6x + x + 3 = 2x² + 7x + 3.
  3. At x = 2, both dimensions are 5 feet. Perimeter is 20 feet and area is 25 square feet.
F20. A rectangular board measures x + 5 centimeters by x + 4 centimeters. A rectangular label measuring x centimeters by x + 2 centimeters covers part of it, where x > 0. Find a simplified polynomial for the uncovered area, classify it, state its degree, and evaluate it when x = 3.
Answer

Uncovered area = 7x + 20 cm², a binomial of degree 1. At x = 3, the uncovered area is 41 cm².

  1. Board area: (x + 5)(x + 4) = x² + 9x + 20. Label area: x(x + 2) = x² + 2x.
  2. Subtract the entire label area: (x² + 9x + 20) − (x² + 2x) = 7x + 20. The quadratic terms cancel.
  3. Two terms remain and the highest exponent is 1. At x = 3: 7(3) + 20 = 41. Check: the board is 8 × 7 = 56 cm² and the label is 3 × 5 = 15 cm²; 56 − 15 = 41.

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