SAT Math — Polynomials and Function Notation

Polynomial Basics

A polynomial is an expression with multiple terms, where the variables have whole-number exponents. You've already seen linear (degree 1) and quadratic (degree 2) polynomials. The SAT also tests higher-degree polynomials.

Key vocabulary:

• Degree: The highest exponent in the polynomial (e.g., x³ + 2x² − x + 5 is degree 3)
• Leading coefficient: The coefficient of the highest-degree term
• Constant term: The term with no variable

Polynomial Operations

Adding/Subtracting: Combine like terms (same variable, same exponent) > (3x² + 5x − 2) + (x² − 3x + 7) = 4x² + 2x + 5

Multiplying: Use the distributive property; multiply every term in one by every term in the other. > (x + 3)(x² − 2x + 1) > = x(x² − 2x + 1) + 3(x² − 2x + 1) > = x³ − 2x² + x + 3x² − 6x + 3 > = x³ + x² − 5x + 3

FOIL (for two binomials): First, Outer, Inner, Last > (x + 4)(x − 2) = x² − 2x + 4x − 8 = x² + 2x − 8

Function Notation: f(x)

f(x) (read "f of x") means a function named f with input x.

Evaluating a function: Substitute the input value wherever you see x. > f(x) = 2x² + 3x − 1 > f(4) = 2(4)² + 3(4) − 1 = 32 + 12 − 1 = 43

f(a + b): Replace every x with (a + b): > f(x) = x + 5 > f(a + b) = (a + b) + 5 = a + b + 5

Composition of Functions

Composite function f(g(x)): Apply g first, then apply f to the result.

> f(x) = 2x + 1 and g(x) = x² > f(g(3)): First, g(3) = 3² = 9. Then f(9) = 2(9) + 1 = 19

> g(f(3)): First, f(3) = 2(3) + 1 = 7. Then g(7) = 7² = 49

Note: f(g(x)) ≠ g(f(x)) in general.

Zeros of a Polynomial

A zero (or root) of a polynomial is a value of x that makes the polynomial equal to zero. For the factored form: > p(x) = (x − 2)(x + 5)(x − 1) > Zeros: x = 2, x = −5, x = 1

The number of zeros (counting multiplicity) equals the degree of the polynomial.

Real-world example: A box is made from a square sheet of cardboard by cutting equal squares of side length x from each corner. If the sheet is 20 inches wide, the volume is V(x) = x(20 − 2x)². A question might ask for the x that maximizes V (test on graphing calculator or by expanding).

---

Key Terms

• Polynomial: An expression with multiple terms with whole-number exponent variables
• Degree: The highest exponent in the polynomial
• f(x) notation: Function notation; f of x; substitute x with the given input
• Composition: f(g(x)) = apply g first, then f; order matters
• Zero/root: An x-value where the polynomial equals 0
• FOIL: Method for multiplying two binomials: First, Outer, Inner, Last
• Like terms: Terms with the same variable and same exponent; can be combined
• Leading coefficient: The coefficient on the highest-degree term

---

Quiz Questions:

Q1. If f(x) = 3x² − x + 2, what is f(−2)?

A) 12 B) 16 C) 8 D) 24

Answer: B — f(−2) = 3(−2)² − (−2) + 2 = 3(4) + 2 + 2 = 12 + 2 + 2 = 16.

---

Q2. Multiply: (x + 3)(x − 7)

A) x² − 21 B) x² − 4x − 21 C) x² + 4x − 21 D) x² − 10x + 21

Answer: B — FOIL: First: x·x = x². Outer: x·(−7) = −7x. Inner: 3·x = 3x. Last: 3·(−7) = −21. Combine: x² − 7x + 3x − 21 = x² − 4x − 21.

---

Q3. Functions f and g are defined as: f(x) = x + 4 and g(x) = 2x. What is f(g(5))?

A) 18 B) 14 C) 9 D) 22

Answer: B — g(5) = 2(5) = 10. Then f(10) = 10 + 4 = 14.

---

Q4. The polynomial p(x) = (x − 1)(x + 3)(x − 5) has which zeros?

A) x = 1, x = −3, x = 5 B) x = −1, x = 3, x = −5 C) x = 1, x = 3, x = 5 D) x = −1, x = −3, x = −5

Answer: A — Set each factor equal to zero: x − 1 = 0 → x = 1; x + 3 = 0 → x = −3; x − 5 = 0 → x = 5.

---

Q5. If h(x) = x² + 1, what is h(a + 2)?

A) a² + 1 B) a² + 4a + 5 C) a² + 2a + 5 D) a² + 4 + 1

Answer: B — Replace every x with (a + 2): h(a + 2) = (a + 2)² + 1 = a² + 4a + 4 + 1 = a² + 4a + 5.