Equivalent Expressions Pattern - Factoring

Factoring Polynomials

Factoring means rewriting a polynomial as a product of simpler expressions. The SAT tests three main factoring techniques: pulling out a greatest common factor (GCF), factoring trinomials, and recognizing a difference of squares. Many harder questions combine these steps or use the Factor Theorem to find unknowns.

The Core Techniques

GCF Factoring: Find the largest factor shared by every term — both the numerical coefficients and the variable parts. Factor it out front.

$5y^2 - 15y + 45 = 5(y^2 - 3y + 9)$ — the GCF is $5$ (not $5y$, because $45$ has no $y$)

Trinomial Factoring ($x^2 + bx + c$): Find two numbers that multiply to $c$ and add to $b$.

$a^2 + 4a - 12$: need two numbers with product $-12$ and sum $+4$ → that's $+6$ and $-2$, so $(a + 6)(a - 2)$

Difference of Squares: $A^2 - B^2 = (A - B)(A + B)$. Both terms must be perfect squares separated by subtraction.

$4a^2 - 81b^2 = (2a)^2 - (9b)^2 = (2a - 9b)(2a + 9b)$

Factoring by Grouping (leading coefficient $\neq 1$): For $ax^2 + bx + c$, find two numbers that multiply to $ac$ and add to $b$, split the middle term, then group.

$2x^2 - 7x - 30$: $ac = -60$, need sum $-7$ → use $-12$ and $5$
$= 2x^2 - 12x + 5x - 30 = 2x(x - 6) + 5(x - 6) = (2x + 5)(x - 6)$

Worked Examples

Example 1. Which expression is a factor of $5y^2 - 15y + 45$?

A) $5y^2$
B) $y$
C) $5$
D) $-15y$

The GCF of the coefficients $5$, $-15$, and $45$ is $5$. Since the constant term $45$ has no variable $y$, the variable is not part of the GCF.
$5y^2 - 15y + 45 = 5(y^2 - 3y + 9)$
Gotcha: Option A ($5y^2$) is the first term, not a factor. Option B ($y$) divides the first two terms but not $45$. A factor must divide every term.
The answer is C.

Example 2. Which expression is equivalent to $15p^9q^3 - 25p^3q^3$?

A) $5p^3q^3(3p^3 - 5)$
B) $5p^3q^3(3p^6 - 5)$
C) $5p^3q^3(15p^6)$
D) $5p^3q^3(3p^3)$

GCF of $15$ and $25$ is $5$. GCF of $p^9$ and $p^3$ is $p^3$ (take the smaller exponent). GCF of $q^3$ and $q^3$ is $q^3$.
Overall GCF: $5p^3q^3$
Factor out: $\dfrac{15p^9q^3}{5p^3q^3} = 3p^{9-3} = 3p^6$ and $\dfrac{25p^3q^3}{5p^3q^3} = 5$
Result: $5p^3q^3(3p^6 - 5)$
Gotcha: Option A divides the exponents ($9 \div 3 = 3$) instead of subtracting ($9 - 3 = 6$). When factoring out a variable, you subtract exponents.
The answer is B.

Example 3. Which expression is equivalent to $a^2 + 4a - 12$?

A) $(a - 6)(a + 2)$
B) $(a + 6)(a - 2)$
C) $(a + 3)(a - 4)$
D) $(a + 12)(a - 1)$

We need two numbers that multiply to $-12$ and add to $+4$.
Try: $+6$ and $-2$ → product $= -12$ ✓, sum $= +4$ ✓
So: $(a + 6)(a - 2)$
Gotcha: Option A reverses the signs — $(a - 6)(a + 2)$ expands to $a^2 - 4a - 12$ (the middle term is $-4a$ instead of $+4a$). The sign of the middle term tells you which factor gets the positive number.
The answer is B.

Example 4. Which expression is equivalent to $5(4a^2 - 81b^2)$?

A) $5(2a - 3b)(2a + 27b)$
B) $5(2a - 9b)^2$
C) $5(2a - 9b)(2a + 9b)$
D) $5(2a - 81b)(2a + b)$

Recognize $4a^2 - 81b^2$ as a difference of squares: $(2a)^2 - (9b)^2$.
Apply the formula: $(2a - 9b)(2a + 9b)$. Keep the $5$ out front.
Result: $5(2a - 9b)(2a + 9b)$
Gotcha: Option B is $(2a - 9b)^2 = 4a^2 - 36ab + 81b^2$ — a perfect square trinomial, not a difference of squares. A difference of squares always has one $+$ factor and one $-$ factor.
The answer is C.

Example 5. Which of the following is a factor of $2x^2 - 7x - 30$? I. $x + 6$ II. $2x + 5$

A) I only
B) II only
C) I and II
D) Neither I nor II

Since the leading coefficient is $2$, use factoring by grouping. Find two numbers that multiply to $2 \times (-30) = -60$ and add to $-7$: those are $-12$ and $5$.
Split the middle term: $2x^2 - 12x + 5x - 30$
Group: $2x(x - 6) + 5(x - 6) = (2x + 5)(x - 6)$
The factors are $(2x + 5)$ and $(x - 6)$. Statement II matches; statement I has the wrong sign ($x + 6$ vs. $x - 6$).
Gotcha: $(x + 6)$ and $(x - 6)$ are different factors. A single sign flip changes everything.
The answer is B.

Example 6. The polynomial $3y^3 + 51y^2 + 180y$ has a factor of $y + c$, where $c$ is a positive constant. What is the largest possible value of $c$?

First, factor out the GCF of all three terms: $3y$.
$3y(y^2 + 17y + 60)$
Now factor the trinomial: find two numbers with product $60$ and sum $17$ → that's $5$ and $12$.
$3y(y + 5)(y + 12)$
The factors of the form $(y + c)$ with positive $c$ are $(y + 5)$ and $(y + 12)$. The largest value of $c$ is $\boxed{12}$.

Example 7. If $z + d$ is a factor of $z^2 - \dfrac{1}{11}gd^2$, where $g$ and $d$ are constants and $d \neq 0$, what is the value of $g$?

A) $11$
B) $-\dfrac{1}{11}$
C) $-11$
D) $\dfrac{1}{11}$

If $z + d$ is a factor, then substituting $z = -d$ makes the expression zero:
$(-d)^2 - \dfrac{1}{11}g \cdot d^2 = 0$
$d^2 - \dfrac{1}{11}gd^2 = 0$
Since $d \neq 0$, divide by $d^2$: $1 - \dfrac{g}{11} = 0$, so $\dfrac{g}{11} = 1$, giving $g = 11$.
Gotcha: Option D ($\dfrac{1}{11}$) is the value of $\dfrac{g}{11}$, not $g$ itself. Don't stop one step too early.
The answer is A.

What to Do on Test Day

• Always start with the GCF. Factor it out first, then look for additional factoring inside the parentheses.
• Trinomial shortcut: For $x^2 + bx + c$, find two numbers with product $c$ and sum $b$. The signs of those numbers match the signs in the factors.
• Difference of squares checklist: (1) Is it a subtraction? (2) Are both terms perfect squares? If yes: $A^2 - B^2 = (A-B)(A+B)$.
• Leading coefficient $\neq 1$: Use factoring by grouping — find two numbers with product $ac$ and sum $b$, split the middle term, then group.
• Factor Theorem: If $(x + c)$ is a factor, then plugging $x = -c$ into the polynomial gives $0$. Use this to find unknown constants.
• Sign errors are the #1 trap. $(a + 6)(a - 2)$ vs. $(a - 6)(a + 2)$ give different middle terms. Always check by expanding.
• GCF of variable terms: Take the smallest exponent, and subtract exponents when dividing: $p^9 \div p^3 = p^{9-3} = p^6$, not $p^{9 \div 3} = p^3$.