Polynomial Review · Lesson 4

The Factoring Toolkit.

Factoring is un-multiplying: rewriting a polynomial as a product of simpler factors. The secret to never getting stuck is to run the tools in the SAME order every time — GCF first, then look at the number of terms — and to check every answer by multiplying back. This is the skill that carries straight into solving quadratics next week.

By the end, you can

  • 1I can factor out the greatest common factor (numbers and variables) and verify by re-distributing.
  • 2I can factor a difference of squares a² − b² = (a+b)(a−b), and recognize when a binomial is one.
  • 3I can factor a trinomial with a = 1 by finding two numbers with product c and sum b.
  • 4I can factor a trinomial with a ≠ 1 by decomposition (splitting the middle term, then grouping).
  • 5I can factor a four-term polynomial by grouping.
  • 6I can factor COMPLETELY — pulling the GCF first and continuing until every factor is prime — and always check by expanding.
01 / CONCEPTS

The core ideas.

The order — run it top to bottom, every time

1) GCF first, always. 2) Count the terms of what remains: 2 terms → try difference of squares; 3 terms → trinomial (a = 1: find two numbers; a ≠ 1: decomposition); 4 terms → grouping. 3) Check by multiplying back. Pulling the GCF first shrinks the numbers and often reveals a clean pattern underneath — skipping it is the #1 reason a trinomial 'won't factor.'

Why it works

A fixed checklist means you never freeze in front of an expression. You always know the next thing to try, and you keep going until nothing factors further.

Greatest Common Factor (GCF)

The GCF is the biggest chunk every term shares — the largest number dividing all coefficients, times the LOWEST power of each common variable. Factor it out by division: 6x³ + 9x² = 3x²(2x + 3).

6x3+9x2=3x2(2x+3)6x^3 + 9x^2 = 3x^2(2x + 3)

Why it works

Factoring out the GCF is the distributive law run backwards: 'what did I hand to every term?' For variables you can only pull out the lowest power present, because a term with fewer x's can only share that many.

Difference of squares (2 terms)

If a binomial is (something)² − (something)², it factors as sum × difference: a² − b² = (a + b)(a − b). Example: 9x² − 16 = (3x + 4)(3x − 4). Two warnings: the terms must be perfect squares AND separated by a minus. A SUM of squares a² + b² does not factor over the real numbers.

a2−b2=(a+b)(a−b)a^2 - b^2 = (a+b)(a-b)

Why it works

This is the special-products identity read right-to-left. Recognizing perfect squares fast — 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 and x², 4x², 9x² — is the whole skill.

Trinomials with a = 1: find two numbers

For x² + bx + c, find two numbers that multiply to c and add to b; those are the constants in the factors. x² + 7x + 12 → 3 and 4 → (x + 3)(x + 4). Signs: if c > 0 the two numbers share b's sign; if c < 0 they have opposite signs (the bigger carries b's sign). x² − x − 12 → 3 and −4 → (x + 3)(x − 4).

x2+(p+q)x+pq=(x+p)(x+q)x^2 + (p+q)x + pq = (x+p)(x+q)

Why it works

Expanding (x+p)(x+q) gives x² + (p+q)x + pq, so b is the SUM and c is the PRODUCT of the mystery numbers. List the factor pairs of c and read off the pair whose sum is b.

Trinomials with a ≠ 1: decomposition

For ax² + bx + c: (1) multiply a·c; (2) find two numbers that multiply to a·c and add to b; (3) split the middle term bx into those two; (4) group and factor each pair — a common binomial appears. Example: 3x² + 10x + 8, a·c = 24, numbers 6 and 4 → 3x² + 6x + 4x + 8 → 3x(x + 2) + 4(x + 2) → (x + 2)(3x + 4).

3x2+10x+8=(x+2)(3x+4)3x^2 + 10x + 8 = (x+2)(3x+4)

Why it works

The a·c trick isn't magic — it rebuilds the two middle cross-terms that FOIL produced, then grouping harvests them back into factors. You are literally reversing binomial multiplication.

Grouping (4 terms) and checking

With four terms, pair them, factor the GCF from each pair, and pull out the shared binomial: x³ + 2x² + 3x + 6 = x²(x + 2) + 3(x + 2) = (x + 2)(x² + 3). Always CHECK by multiplying the factors back out — factoring and expanding are inverses, so re-multiplying must return the original.

x3+2x2+3x+6=(x+2)(x2+3)x^3 + 2x^2 + 3x + 6 = (x+2)(x^2+3)

Why it works

Grouping is GCF applied locally to each pair, hoping both reveal the SAME parenthesis. If they don't match, re-pair before giving up. Decomposition even ends in a grouping step — same tool.

02 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Factor 10x² + 15x.

1

Find the GCF

GCF of 10 and 15 is 5; both terms have x → GCF = 5x.

2

Divide it out

5x(2x + 3).

3

Check

5x·2x + 5x·3 = 10x² + 15x. ✓

EX 2easy

Factor x² − 36.

1

Recognize the pattern

x² and 36 = 6² are perfect squares separated by a minus → difference of squares.

EX 3easy

Factor x² + 9x + 20.

1

Product and sum

Need two numbers with product 20 and sum 9 → 4 and 5.

2

Write the factors

(x + 4)(x + 5). Check the middle: 4x + 5x = 9x. ✓

EX 4medium

Factor x² − 2x − 15.

1

Product and sum

Product −15, sum −2. Since c < 0, opposite signs → 3 and −5 (bigger one carries the minus).

2

Write and check

(x + 3)(x − 5). Middle: 3x − 5x = −2x; last: 3·(−5) = −15. ✓

EX 5medium

Factor 2x² − 50.

1

GCF first

2(x² − 25).

2

Now the bracket is a difference of squares

x² − 25 = (x + 5)(x − 5).

EX 6medium

Factor 2x² + 7x + 3 by decomposition.

1

a·c and the two numbers

a·c = 6; product 6, sum 7 → 6 and 1.

2

Split the middle

2x² + 6x + x + 3.

3

Group each pair

2x(x + 3) + 1(x + 3) → (x + 3)(2x + 1).

EX 7challenge

Factor completely 4x³ − 24x² + 36x.

1

GCF first

GCF of 4, 24, 36 is 4; lowest power of x is x → GCF = 4x. This gives 4x(x² − 6x + 9).

2

Factor the trinomial

Product 9, sum −6 → −3 and −3, so x² − 6x + 9 = (x − 3)².

3

Check

4x(x − 3)² = 4x(x² − 6x + 9) = 4x³ − 24x² + 36x. ✓

03 / COMMON TRAPS

Where students slip.

Trap

Skipping the GCF and declaring a trinomial 'prime.'

Fix

Always pull the GCF first. 2x² − 50 looks stuck until you factor the 2: 2(x² − 25) = 2(x + 5)(x − 5). The common factor was in the way the whole time.

Trap

Trying to factor a SUM of squares: x² + 9 = (x + 3)(x + 3)?

Fix

No. (x + 3)² = x² + 6x + 9, not x² + 9. A sum of squares does not factor over the real numbers — leave it as is.

Trap

Getting the signs wrong in a trinomial.

Fix

Use the rule: c > 0 → same signs (both match b); c < 0 → opposite signs (larger carries b's sign). Then CHECK by re-multiplying.

Trap

Stopping too early — leaving 2(x² − 25) or (2x − 6)(x − 3).

Fix

'Factor completely' means every factor is prime. Finish: 2(x² − 25) = 2(x + 5)(x − 5); and (2x − 6) still has a GCF of 2. Keep going until nothing factors.

Trap

Concluding a four-term polynomial 'won't group' after one attempt.

Fix

If the two binomials don't match, re-pair the terms (swap the order of the middle two) before giving up.

04 / QUIZ

Test yourself.

The Test

14 problems across three tiers. Auto-graded on submit. Hints and a full walkthrough on every question. Your work is saved on this device — sign in to keep it across devices.

0/14Answered
How to answerType just the value. Formatting won't trip you up — we ignore spaces, capital letters, commas, and subscript style, so t₁ = 5, d = 3 and t1=5,d=3 both count. For a list of numbers, separate them with commas (e.g. 3, 10, 17).
Core

Core (6)

C1

Factor 10x² + 15x.

Need a hint?
  • Take the numeric GCF of 10 and 15, and the shared x.
  • GCF = 5x.
C2

Factor x² − 49.

Need a hint?
  • Two perfect squares, minus sign → sum × difference.
C3

Factor x² + 6x + 8.

Need a hint?
  • Two numbers with product 8 and sum 6.
C4

Factor x² + 3x − 10.

Need a hint?
  • Product −10, sum 3 → opposite signs, bigger carries the +.
C5

Factor x² − 9x + 18.

Need a hint?
  • Product 18, sum −9 → both negative.
C6

When factoring any polynomial, what should you always try FIRST?

Need a hint?
  • The step that shrinks the numbers and reveals hidden patterns.
Apply

Apply (5)

A1

Factor completely: 3x² − 27.

Need a hint?
  • GCF first, then look at the bracket.
  • 3(x² − 9), and x² − 9 is a difference of squares.
A2

Factor 2x² + 11x + 12.

Need a hint?
  • a·c = 24; find product 24, sum 11.
  • 8 and 3: split 2x² + 8x + 3x + 12.
A3

Factor by grouping: x³ + 4x² + 2x + 8.

Need a hint?
  • Pair the terms and pull a GCF from each pair.
  • x²(x + 4) + 2(x + 4).
A4

Factor 9x² − 25.

Need a hint?
  • 9x² = (3x)² and 25 = 5².
A5

Which expression does NOT factor over the real numbers?

Need a hint?
  • A SUM of squares has no real factorization.
Extension

Extension (3)

E1

Factor completely: 5x³ − 45x.

Need a hint?
  • GCF 5x first, then the bracket is a difference of squares.
  • 5x(x² − 9).
E2

Factor completely: 2x² − 12x + 18.

Need a hint?
  • GCF 2 first; the bracket is a perfect-square trinomial.
  • 2(x² − 6x + 9), and x² − 6x + 9 = (x − 3)².
E3

Factor completely: 4x³ − 24x² + 36x.

Need a hint?
  • GCF 4x first, then factor the trinomial.
  • 4x(x² − 6x + 9), and product 9 sum −6 → (x − 3)².