Pre-Calculus · Polynomial Expansion
Pre-Calc Diagnostic: Expand and Simplify (Stewart Problem 3)
Five expansions from Stewart's Pre-Calculus Diagnostic Test — the distributive property, FOIL, conjugate radical products, the perfect-square binomial, and cubing a binomial. The five patterns that show up in every AP Calculus problem, taught the way we teach them at SOMATH on the Upper West Side.
Direct answers:
(a) 3(x + 6) + 4(2x − 5) = 11x − 2
(b) (x + 3)(4x − 5) = 4x2 + 7x − 15
(c) (a + b)(a − b) = a − b
(d) (2x + 3)2 = 4x2 + 12x + 9
(e) (x + 2)3 = x3 + 6x2 + 12x + 8
These five patterns are the polynomial algebra reflexes pre-calculus is supposed to drill before AP Calculus arrives. The diagnostic isn't testing if you can do them — it's testing if you can do them fast, in the right order, without losing a sign or forgetting a middle term. Let's walk through each.
Why Problem 3 matters
Every AP Calculus problem buries a polynomial expansion inside it. To differentiate (2x + 3)2 you can use the chain rule, but you can also expand to 4x2 + 12x + 9 and differentiate term-by-term. To integrate (x + 2)3, expansion is sometimes the cleanest route. The five patterns in Problem 3 — distributive, FOIL, difference of squares, perfect square, and cube of a binomial — are the cumulative algebra grammar AP Calc assumes you own.
If any one of them is shaky, you'll be slow in calculus. If two are shaky, you'll be wrong in calculus. Let's drill each.
(a) 3(x + 6) + 4(2x − 5) — distributive property and combining like terms
Two coefficients to distribute, two parentheses to clear, then combine. Nothing fancy — just sign discipline.
Step 1: distribute each coefficient through its parentheses:
3(x + 6) = 3x + 18
4(2x − 5) = 8x − 20
Step 2: combine like terms — group the x terms and the constants:
3x + 18 + 8x − 20 = (3x + 8x) + (18 − 20) = 11x − 2
The trap students fall into: forgetting to distribute the 4 to the −5 and writing 8x + 5 or 8x − 5. The minus sign belongs to the 5, not just the second term. Distribute carefully, then combine.
(b) (x + 3)(4x − 5) — FOIL
FOIL stands for First, Outer, Inner, Last — the four products you make when multiplying two binomials. It's a mnemonic for the distributive property applied twice.
FOIL:
- First: x · 4x = 4x2
- Outer: x · (−5) = −5x
- Inner: 3 · 4x = 12x
- Last: 3 · (−5) = −15
Combine the middle (Outer + Inner) terms:
4x2 − 5x + 12x − 15 = 4x2 + 7x − 15
Stewart's textbook shows the alternative method too — distributing each term: (x + 3)(4x − 5) = x(4x − 5) + 3(4x − 5). Both give the same answer. FOIL is faster on pure binomials; the distribute-each-term method generalizes to trinomials and beyond.
(c) (a + b)(a − b) — the conjugate radical product
This is the difference-of-squares pattern in radical form. Memorize the pattern; you'll use it constantly when rationalizing denominators in pre-calc and calculus.
The difference-of-squares pattern:
(P + Q)(P − Q) = P2 − Q2
With P = a and Q = b:
(a + b)(a − b) = (a)2 − (b)2 = a − b
The square root and the square cancel exactly — that's what makes this pattern useful. The conjugate (a − b) kills the radical when multiplied against (a + b). We'll use this every time we rationalize a denominator like 15 + 2 later in the year.
(d) (2x + 3)2 — the perfect-square binomial
The single most common pre-calc mistake in the country: writing (2x + 3)2 = 4x2 + 9. Wrong. You can't just square each term. Squaring distributes the way multiplication distributes — through both terms, both ways.
The perfect-square pattern:
(a + b)2 = a2 + 2ab + b2
The middle term 2ab is what students forget. With a = 2x and b = 3:
(2x)2 + 2 · (2x) · 3 + 32 = 4x2 + 12x + 9
Verify via FOIL: (2x + 3)(2x + 3) — F: 4x2, O: 6x, I: 6x, L: 9. Sum: 4x2 + 12x + 9. Same answer.
This is the pattern behind completing the square in pre-calc and Algebra II. Lock it in.
(e) (x + 2)3 — cubing a binomial
Cubing is three factors multiplied: (x + 2)(x + 2)(x + 2). We don't memorize a special pattern for this one. We multiply pairs, then carry forward.
Step 1: multiply the first two factors using the perfect-square pattern:
(x + 2)(x + 2) = x2 + 4x + 4
Step 2: multiply that result by the third factor (x + 2) using distribute-each-term:
(x2 + 4x + 4)(x + 2) = x2(x + 2) + 4x(x + 2) + 4(x + 2)
= x3 + 2x2 + 4x2 + 8x + 4x + 8
Step 3: combine like terms:
x3 + (2x2 + 4x2) + (8x + 4x) + 8 = x3 + 6x2 + 12x + 8
The general cube-of-a-binomial pattern is worth memorizing too:
(a + b)3 = a3 + 3a2b + 3ab2 + b3
With a = x and b = 2: x3 + 3x2(2) + 3x(22) + 23 = x3 + 6x2 + 12x + 8. Same answer. Two paths, one destination — pick whichever is faster for you.
The five patterns every pre-calc student should own cold
Problem 3 is a cumulative drill of the five most-used polynomial expansions in calculus:
- Distributive: k(a + b) = ka + kb
- FOIL / distributive twice: (a + b)(c + d) = ac + ad + bc + bd
- Difference of squares: (a + b)(a − b) = a2 − b2
- Perfect-square binomial: (a + b)2 = a2 + 2ab + b2
- Cube of a binomial: (a + b)3 = a3 + 3a2b + 3ab2 + b3
The student who can apply any of these in under 30 seconds — without writing FOIL on the page first — is the student who finishes the Stewart diagnostic in 20 minutes instead of 60. That speed gap is the difference between calculus feeling hard and calculus feeling like algebra with new vocabulary.
How we teach this at SOMATH
Polynomial expansion is one of the first units we drill with pre-calc and incoming AP Calc students. Our sequence in the first six sessions:
- Pure distributive drills. Twenty problems like part (a) in one sitting, with negative coefficients and missing constants. The goal is reflex distribution.
- FOIL drills, then beyond. Twenty binomial-times-binomial problems, then twenty trinomial-times-binomial, where FOIL stops working and the distribute-each-term method takes over.
- The four named patterns. Difference of squares, perfect square, sum of cubes, difference of cubes. Each with twenty mixed problems where the student has to recognize which pattern applies before applying it.
- Reverse direction — factoring. Once expansion is reflex, we run all four patterns backwards. 4x2 + 12x + 9 should jump off the page as (2x + 3)2 in under three seconds.
By the end of the unit, students are FOILing in their heads on the way home from class. That's the bar.
Pre-Calculus tutoring on the Upper West Side?
School of Math (SOMATH) is a small-group math tutoring program at 226 W 79th St — Pre-Calculus, AP Calculus AB / BC, AP Statistics, Algebra I, Algebra II, Geometry, SHSAT prep, SAT math, and Regents prep. Our pre-calc students walk into AP Calculus with the algebra reflexes the course assumes. We start every student with a free 30-minute evaluation and deliver a written diagnostic within 48 hours, even if you don't enroll.
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FAQ
How do you expand 3(x + 6) + 4(2x − 5)?
Distribute each coefficient into its parentheses, then combine like terms. 3(x + 6) = 3x + 18 and 4(2x − 5) = 8x − 20. Combining: 3x + 8x + 18 − 20 = 11x − 2.
What is FOIL and how do you use it on (x + 3)(4x − 5)?
FOIL = First, Outer, Inner, Last — the four products when multiplying two binomials. (x)(4x) + (x)(−5) + (3)(4x) + (3)(−5) = 4x2 − 5x + 12x − 15 = 4x2 + 7x − 15.
Why does (a + b)(a − b) simplify to a − b?
It's the difference-of-squares pattern (P + Q)(P − Q) = P2 − Q2 with P = a and Q = b. So (a)2 − (b)2 = a − b. The middle FOIL terms cancel because they're opposites.
What's the perfect-square pattern for (2x + 3)2?
(a + b)2 = a2 + 2ab + b2. With a = 2x, b = 3: (2x)2 + 2(2x)(3) + 32 = 4x2 + 12x + 9. The middle term 2ab is what most students forget.
How do you cube a binomial like (x + 2)3?
Two paths. Path 1: multiply (x + 2)(x + 2) = x2 + 4x + 4, then multiply by (x + 2) again. Path 2: use the pattern (a + b)3 = a3 + 3a2b + 3ab2 + b3. Either way: x3 + 6x2 + 12x + 8.
Do you offer pre-calculus tutoring on the Upper West Side?
Yes — pre-calculus and AP Calc AB/BC are core to our program. Most of our AP Calc students take pre-calc with us the year before. Our entire program operates from 226 W 79th St on the Upper West Side.