Polynomials and zeros — Digital SAT Math practice
Factors, remainders, zeros and polynomial identities.
Key terms
- Zero: An input value that makes the polynomial output zero; corresponds to an x-intercept on the graph.
- Degree: The highest exponent in a polynomial; limits the maximum number of real zeros and determines end behavior.
- Leading Coefficient: The number multiplying the term with the highest power; dictates whether the ends point up or down.
- Multiplicity: How many times a specific zero appears as a factor; determines if the graph crosses or touches the axis.
Common traps
- Assuming every polynomial has real zeros; odd-degree polynomials always have at least one, but even-degree ones may not cross the x-axis.
- Confusing the zero with the factor; if p(3) = 0, the zero is 3 but the factor is (x - 3), not (x + 3).
- Forgetting that multiplicity affects graph behavior; a zero with even multiplicity touches the axis while odd multiplicity crosses it.
Key ideas to know
Factor Theorem Application
If x − 3 is a factor of p(x), what must be true about p(3)?
p(3) must equal 0. The Factor Theorem states that x − r is a factor exactly when p(r) = 0.
Watch out: Assuming p(r) = 1 or any nonzero value means x − r is still a factor.
Using the Remainder Theorem
How can you quickly find the remainder when p(x) = 2x3 − 4x + 1 is divided by x + 2 without long division?
Evaluate p(−2). The Remainder Theorem says dividing by x − r leaves a remainder of p(r). Here r = −2, so calculate p(−2).
Watch out: Forgetting to flip the sign of r when it is written as x + k; use r = −k, not r = k.
Factoring Differences of Squares
Factor the expression 9x2 − 16 completely over the real numbers.
(3x − 4)(3x + 4). Use a2 − b2 = (a − b)(a + b) where a = 3x and b = 4.
Watch out: Writing (3x − 4)2 or (9x − 16)(1), which incorrectly treats it as a perfect square or fails to factor the squares first.
Polynomial Zeros and End Behavior
A polynomial p(x) has degree 5 and a positive leading coefficient. What is the maximum number of real zeros it can have, and how does it behave as x approaches infinity?
It can have at most 5 real zeros. As x approaches infinity, p(x) approaches positive infinity because the end behavior follows the leading term.
Watch out: Confusing the degree with the exact number of zeros; a degree n polynomial has AT MOST n real zeros, not exactly n.
Zeros vs Intercepts Confusion
A student says the zero of f(x) = x2 − 4 is the point (0, −4). What is wrong with this statement and what is correct?
The zero is the input that makes the output 0, not the y-intercept. The zero is x = 2 (and x = −2). The point (0, −4) is the y-intercept.
Watch out: Equating a zero directly with an ordered pair or confusing it with where the graph crosses the y-axis.
Sample questions
Question 1
The expression (x + 1)(x − 2) is equivalent to x² + bx + c, where b and c are constants. What is the value of b?
- −2
- 3
- 1
- −1
Show answer and explanation
Answer: −1
Expanding: x² + (1 − 2)x + (1)(−2). So b = −1 and c = −2.
Question 2
The function f(x) = x² + bx + 12, where b is a constant, satisfies f(2) = 0. What is the other zero of f?
- −6
- 2
- 4
- 6
Show answer and explanation
Answer: 6
f(2) = 4 + 2b + 12 = 0 → b = −8. Then x² − 8x + 12 = (x − 2)(x − 6), so the other zero is 6. (Shortcut: product of zeros = 12, so other zero = 12/2 = 6.)