Algebra 1 Class 11: Polynomials — Adding, Subtracting, Multiplying — 10 SOMATH Posters, Theory & 50 Practice Questions

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, x^1/2) are not polynomials.

Why it matters

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

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.

The three named types

Simplify first

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

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.

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.

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.

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.

Two-step rule

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

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.

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.