The laws of exponents as counting factors, including zero and negative exponents; polynomials, their degree and standard form; adding and subtracting without losing a sign; multiplying with the area model, and the two special products every test expects you to know by heart. Then factoring as multiplication run backwards: the GCF, a difference of squares, a perfect-square trinomial, x² + bx + c with any signs, and the ac method when the leading coefficient is not 1. Study cards, hints, a practice quiz at three levels, and four workshop activities that check themselves.
A square whose side is x + 4. Its area is (x + 4)² — that is what squaring means.
(x + 4)² = x² + 8x + 16. Squaring a binomial means multiplying it by itself, box by box — and the two matching boxes are why there is always a middle term.
What people get wrong
⚠️People often think…
(x + 4)² is x² + 16.
You cannot square the pieces separately. (x + 4)² is (x + 4)(x + 4) = x² + 8x + 16. Test it with a number: x = 1 gives 25, while x² + 16 gives 17.
The middle term is always there. Test with x = 1.
⚠️People often think…
x³ · x⁴ = x¹².
Same base multiplied means you ADD: x⁷. Three x's next to four x's is seven x's. It is (x³)⁴ that multiplies to x¹², because that is four copies of a group of three.
Parentheses multiply. A dot adds.
⚠️People often think…
(5x − 3) − (2x + 4) is 3x + 1.
The minus sign applies to everything in the parentheses, including the +4. It becomes 5x − 3 − 2x − 4 = 3x − 7. One forgotten sign is the whole difference between 1 and −7.
Distribute the minus before you combine anything.
⚠️People often think…
5x² − 20x factors to 5(x² − 4x), and that is finished.
Both terms also share an x, so the greatest common factor is 5x, not 5. Finished means 5x(x − 4). “Factor completely” means keep pulling until nothing is left in common.
Pull out the number AND the variable.
Worth knowing cold
Every exponent law, and why
All five are the same idea: an exponent counts factors, so the laws just count them for you.
xᵃ · xᵇAdd. Three x's beside four x's is seven x's.
xᵃ ÷ xᵇSubtract. Cancel the matching ones and count what is left.
(xᵃ)ᵇMultiply. It is b copies of a group of a.
x⁰Always 1. Divide anything by itself and see.
x⁻ᵃFlip it: 1/xᵃ. A small positive number, never a negative one.
FactoringGCF first, every single time — number and variable both.
SignsIn x² + bx + c, the bigger factor takes the sign of b.
Box by box, sign by sign
1Watch one
Expand (x + 4)².
It means (x + 4)(x + 4).
Four boxes: x², 4x, 4x, 16.
The two matching boxes add: 8x.
x² + 8x + 16.
2Do one with me
Simplify (5x − 3) − (2x + 4). Watch the minus.
The minus hits both terms. +4 becomes
Combine the x terms: 5x − 2x =
Combine the numbers: −3 − 4 =
💬One sentence, then you move on
Why is there always a middle term when you square a binomial?
3Try one
Factor 5x² − 20x completely.
I want a hint first
5x² and 20x both contain a 5 and at least one x. Pull out everything they share, then check by multiplying back.
💬Last one — then you're done here
How do you know when a factoring problem is actually finished?
Where this goes
Where this lives
Anywhere a quantity grows by a multiplier instead of a fixed amount: compound interest, a population, a signal getting weaker through a wall.
What this feeds
Next unit is quadratics, where factoring stops being practice and starts being how you solve.
Write one expression you could factor, and say what comes out of both terms.
One card at a time — tap “Show me” to check yourself, then Next. Start at Foundation; when those feel easy, climb.
Helpful Hints
🔍 Factoring any polynomial — the routine
1 · Pull out the GCF first: 3x² + 15x + 18 = 3(x² + 5x + 6). 2 · Count the terms. Two terms: is it a² − b²? Three terms: is it a perfect square (middle = 2 · √first · √last)? If not, hunt for two numbers with product a · c and sum b. 3 · Split, pair, pull out the shared binomial when a ≠ 1. 4 · Multiply your answer back out. 3(x + 2)(x + 3) = 3(x² + 5x + 6) = 3x² + 15x + 18 ✓.
GCF, count, pattern, check.
🟫 The whole unit in one picture — the area model
A product of two polynomials is a rectangle cut into cells: (x + 3)(x + 5) is x², 3x, 5x and 15, so x² + 8x + 15. Multiplying fills the cells from the sides; factoring reads the sides from the cells. Exponent laws are the same idea one level down: x³ · x⁴ is seven x’s in a row, so x⁷. Everything here is counting factors and adding up cells.
Sides give cells; cells give sides.
⚠️ Traps the test loves
x³ · x⁴ = x¹². No: same base multiplied ADDS exponents, x⁷. Only (x³)⁴ multiplies them, x¹².
2⁻³ = −8. No: a negative exponent flips, 1/8. It never makes the value negative.
(x + 4)² = x² + 16. No: it is (x + 4)(x + 4) = x² + 8x + 16. The middle term is always there.
(5x − 3) − (2x + 4) = 3x + 1. No: the minus hits the 4 too, 3x − 7.
5x² − 20x = 5(x² − 4x) and stopping. Not complete — the GCF is 5x, so 5x(x − 4).
Factoring x² − 2x − 15 as (x + 5)(x − 3). Signs: the BIGGER number takes the sign of b, so (x − 5)(x + 3).
📋 The four factor patterns — the chart
Pattern
How to spot it
Example
GCF
every term shares a number or a power of x
6x² + 9x = 3x(2x + 3)
Difference of squares
two perfect squares, a minus, no middle term
4x² − 25 = (2x + 5)(2x − 5)
Perfect-square trinomial
first and last are squares; middle = 2 · their roots
x² − 10x + 25 = (x − 5)²
Trinomial, a = 1
two numbers: product c, sum b
x² + 7x + 12 = (x + 3)(x + 4)
Trinomial, a ≠ 1 (ac method)
two numbers: product a · c, sum b; split and pair
2x² + 7x + 3 = (2x + 1)(x + 3)
🎯 How the test will ask
“Simplify” with exponents — a product, a quotient, a power of a power, a zero or negative exponent. Decide which law BEFORE touching the numbers.
“Add / subtract” two polynomials in parentheses. Distribute the minus to every term, then combine like terms only.
“Multiply” two binomials, or a binomial and a trinomial. Four cells or six; combine the middle.
“Factor completely.” GCF first, then the pattern, then check by multiplying. Completely means look again.
“Which expression is equivalent to…” — the answer is a rewritten form. Multiply the choices back out or test a number.
A rectangle whose area is a trinomial — “find the side lengths.” Factor; the factors are the sides.
“What does the factored form show?” The zeros — each factor (x − k) gives the zero x = k.
✅ Before the test, can you…
State all five exponent laws with an example each, including x⁰ = 1 and x⁻ⁿ = 1/xⁿ?
Name the degree, the leading coefficient and the number of terms of a polynomial in standard form?
Subtract two polynomials without losing a sign?
Draw the area model for (x + a)(x + b) and read the product from it?
Expand (a + b)² and (a + b)(a − b) from memory, and explain why (x + y)² is not x² + y²?
Factor a GCF, a difference of squares, a perfect-square trinomial, and x² + bx + c with any signs?
Factor 2x² + 7x + 3 by the ac method and prove it by multiplying?
Pick your level
Look back at anything you missed — the hint that appeared is exactly what to reread tonight.
How sure did you feel?
Workshop
Work like an algebraist: fill the four cells of an area model, put the ac method in order, sort twelve expressions by the pattern that factors them, then settle the oldest argument in algebra — whether (x + y)² is x² + y². Every activity checks itself, and hints are free.
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