Polynomial Operations and Factoring (SAT)

Adding and Subtracting Polynomials

Rule: Combine Like Terms

Like terms: Same variable(s) with same exponent(s)

Example: $(3x^2 + 5x - 2) + (2x^2 - 3x + 7)$ $= 3x^2 + 2x^2 + 5x - 3x - 2 + 7$ $= 5x^2 + 2x + 5$

Subtraction: Distribute the negative! $(4x^2 + 3x) - (2x^2 - x)$ $= 4x^2 + 3x - 2x^2 + x$ $= 2x^2 + 4x$

Multiplying Polynomials

Monomial × Polynomial

Distribute the monomial to each term

$3x(2x^2 - 5x + 1)$ $= 6x^3 - 15x^2 + 3x$

Binomial × Binomial (FOIL)

First, Outer, Inner, Last

$(x + 3)(x + 5)$ $= x^2 + 5x + 3x + 15$ $= x^2 + 8x + 15$

Special Products

Square of a sum: $(a + b)^2 = a^2 + 2ab + b^2$

Square of a difference: $(a - b)^2 = a^2 - 2ab + b^2$

Difference of squares: $(a + b)(a - b) = a^2 - b^2$

Examples: $(x + 4)^2 = x^2 + 8x + 16$ $(x - 3)^2 = x^2 - 6x + 9$ $(x + 5)(x - 5) = x^2 - 25$

Factoring

Greatest Common Factor (GCF)

Pull out what's common

$6x^3 + 9x^2 = 3x^2(2x + 3)$

Factoring Trinomials

$x^2 + bx + c$

Find two numbers that:

• Multiply to $c$
• Add to $b$

Example: $x^2 + 7x + 12$

• Need: multiply to 12, add to 7
• Numbers: 3 and 4
• Factor: $(x + 3)(x + 4)$

Difference of Squares

$a^2 - b^2 = (a+b)(a-b)$

Example: $x^2 - 16 = (x+4)(x-4)$

Perfect Square Trinomials

$a^2 + 2ab + b^2 = (a + b)^2$ $a^2 - 2ab + b^2 = (a - b)^2$

Example: $x^2 + 10x + 25 = (x + 5)^2$

Factoring by Grouping

For 4 terms

$ax + ay + bx + by$ $= a(x+y) + b(x+y)$ $= (a+b)(x+y)$

Polynomial Division

Long Division (rarely on SAT)

Similar to numeric long division

Synthetic Division (rarely on SAT)

Shortcut for dividing by $(x - a)$

SAT Focus: Usually simpler - factor and cancel

SAT Question Types

Type 1: Expand/Multiply

"What is $(2x - 3)(x + 4)$?"

Use FOIL or distribution

Type 2: Factor Completely

"Factor: $x^2 - 9$"

Recognize patterns (difference of squares)

Type 3: Simplify Expressions

"Simplify: $(x^2 + 3x) - (2x^2 - x)$"

Combine like terms

Type 4: Application

"The area of a rectangle is $x^2 + 7x + 12$. If the length is $x + 4$, what is the width?"

Factor and divide: $(x+3)$

SAT Strategies

Recognize Patterns

• Difference of squares: $a^2 - b^2$
• Perfect squares: $a^2 ± 2ab + b^2$

Check by FOIL-ing Back

Factor: $x^2 + 5x + 6 = (x+2)(x+3)$ Check: $(x+2)(x+3) = x^2 + 5x + 6$ ✓

Use Answer Choices

If asked to factor, check answers by multiplying

Common Coefficients

Always check for GCF first!

Common SAT Traps

Trap 1: Sign Errors

$(x - 3)^2 ≠ x^2 - 9$ Correct: $(x-3)^2 = x^2 - 6x + 9$

Trap 2: Incomplete Factoring

$2x^2 + 8x = 2x(x + 4)$ ← Must pull out GCF

Trap 3: Distributing Incorrectly

$(x + 3)^2 ≠ x^2 + 9$ Must include middle term: $x^2 + 6x + 9$

Trap 4: Subtracting Without Parentheses

$(3x) - (2x - 5) = 3x - 2x + 5 = x + 5$ ← Distribute negative!

SAT Tips

• FOIL for binomial multiplication
• Difference of squares appears often! $a^2 - b^2 = (a+b)(a-b)$
• Check GCF first when factoring
• $(a+b)^2 = a^2 + 2ab + b^2$ (don't forget middle term!)
• Verify by expanding factored form
• Use calculator to check numeric results