PSAT/NMSQT Math: Polynomial Operations and Zeros
Polynomials show up throughout PSAT/NMSQT Advanced Math, and this lesson builds the core toolkit for working with them: combining and multiplying polynomials, evaluating them at a specific value, dividing them, and finding their zeros and factors. This free, complete lesson covers adding, subtracting, and multiplying polynomials, evaluating polynomials and the Remainder Theorem, dividing polynomials, and finding zeros and factors. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback and full explanations. Everything here is free, and you can keep practicing afterward with the linked quizzes below.
1. Adding, Subtracting, and Multiplying Polynomials
To add or subtract polynomials, combine like terms (terms with the exact same variable and exponent). When subtracting, distribute the negative sign across every term of the polynomial being subtracted. To multiply polynomials, distribute every term in the first polynomial across every term in the second, then combine like terms.
Worked Example: Subtract: (2x² + x − 5) − (x² − 3x + 2).
Distribute the negative sign: 2x² + x − 5 − x² + 3x − 2.
Combine like terms: x² + 4x − 7
1. Add: (3x² − 2x + 7) + (x² + 5x − 4)
Explanation: Combine like terms: (3x² + x²) + (−2x + 5x) + (7 − 4) = 4x² + 3x + 3.
2. Subtract: (2x² + x − 5) − (x² − 3x + 2) (see worked example above)
Explanation: Distributing the negative sign and combining like terms gives x² + 4x − 7.
3. Expand (x + 3)(2x² − x + 4) and identify the coefficient of x.
Explanation: Expanding: 2x³ − x² + 4x + 6x² − 3x + 12 = 2x³ + 5x² + x + 12. The coefficient of x (not x² or x³) is 1.
2. Evaluating Polynomials and the Remainder Theorem
To evaluate a polynomial P(x) at a specific value, substitute that value for x and simplify. The Remainder Theorem states that dividing P(x) by (x − a) leaves a remainder equal to P(a), meaning you can find a division remainder just by evaluating, with no actual division needed.
Worked Example: Find the remainder when P(x) = 3x³ − 4x + 1 is divided by x − 2.
By the Remainder Theorem, the remainder equals P(2):
P(2) = 3(8) − 4(2) + 1 = 24 − 8 + 1 = 17
1. If P(x) = x³ − 2x² + x − 4, find P(2).
Explanation: P(2) = 8 − 2(4) + 2 − 4 = 8 − 8 + 2 − 4 = −2.
2. Find the remainder when P(x) = 3x³ − 4x + 1 is divided by x − 2. (see worked example above)
Explanation: By the Remainder Theorem, the remainder equals P(2) = 3(8) − 4(2) + 1 = 17.
3. Dividing P(x) by (x + 4) gives a remainder of P(−4). If P(−4) = 0, what does this tell you?
Explanation: A remainder of 0 means the division is exact, so (x + 4) divides evenly into P(x), making it a factor.
3. Dividing Polynomials
Dividing a polynomial by a monomial means dividing each term separately. Dividing by a binomial (like x − a) typically uses long division or synthetic division, matching the process of long division with numbers.
Worked Example: Divide (6x³ − 9x² + 3x) by 3x.
6x³/3x − 9x²/3x + 3x/3x = 2x² − 3x + 1
1. Divide (6x³ − 9x² + 3x) by 3x. (see worked example above)
Explanation: Dividing each term by 3x separately: 6x³/3x = 2x², −9x²/3x = −3x, 3x/3x = 1, giving 2x² − 3x + 1.
2. Divide (x³ − 3x² + 2x) by x, and identify the resulting expression.
Explanation: Dividing each term by x: x³/x − 3x²/x + 2x/x = x² − 3x + 2.
3. When a cubic polynomial is divided by a linear binomial (x − a), what is the degree of the resulting quotient?
Explanation: Dividing a degree-n polynomial by a degree-1 binomial always produces a quotient of degree n − 1, so a cubic (degree 3) divided by a linear term gives a quadratic (degree 2) quotient.
4. Zeros, Factors, and Constructing Polynomials
A zero of a polynomial P(x) is any value a where P(a) = 0; equivalently, (x − a) is a factor of P(x). To construct a polynomial from its zeros, multiply together a factor for each zero, along with the desired leading coefficient.
Worked Example: Which value is a zero of P(x) = x³ − 4x² − x + 4?
Test x = 1: P(1) = 1 − 4 − 1 + 4 = 0.
Since P(1) = 0, x = 1 is a zero
1. Which value is a zero of P(x) = x³ − 4x² − x + 4? (see worked example above)
Explanation: P(1) = 1 − 4 − 1 + 4 = 0, confirming x = 1 is a zero.
2. A polynomial has leading coefficient 2 and zeros at x = 2 and x = −3. What is its constant term?
Explanation: P(x) = 2(x − 2)(x + 3). The constant term comes from 2 · (−2) · 3 = −12.
3. Is (x + 4) a factor of x² − 5x − 24?
Explanation: Testing x = −4: (−4)² − 5(−4) − 24 = 16 + 20 − 24 = 12 ≠ 0, so (x + 4) is not a factor.
Common Mistakes to Avoid
- Forgetting to distribute a negative sign across every term when subtracting one polynomial from another.
- Only multiplying some of the terms when expanding two polynomials. Every term in the first factor must be multiplied by every term in the second.
- Trying to use the Remainder Theorem with the wrong sign. Dividing by (x − a) gives remainder P(a); dividing by (x + a) means a is negative, so you evaluate P(−a).
- Confusing a zero of a polynomial with a factor. If a is a zero (P(a) = 0), then (x − a), not (x + a) or (x − a) with the wrong sign, is the corresponding factor.
- Assuming every candidate value is automatically a zero without checking. Always substitute and confirm the result actually equals 0 before concluding a value is a zero.
Frequently Asked Questions
What's the fastest way to find the remainder of a polynomial division on the PSAT?
Use the Remainder Theorem: to find the remainder when P(x) is divided by (x − a), simply evaluate P(a). No actual long division is required.
How are a polynomial's zeros related to its factors?
Every zero a of a polynomial corresponds to a factor (x − a), and every factor (x − a) corresponds to a zero at x = a. They're two ways of describing the same information.
Where can I practice more problems like these?
The Polynomial Operations and Zeros quizzes in the PSAT/NMSQT Math Question Bank include additional original problems on this topic, along with quizzes covering every other Heart of Algebra, Advanced Math, Problem Solving and Data Analysis, and Geometry and Trigonometry skill tested on the PSAT/NMSQT. You can also browse the full PSAT/NMSQT Math Question Bank.