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Polynomials for the PSAT/NMSQT: Complete Study Guide + Free Practice Problems
Everything the PSAT/NMSQT tests about polynomials in one place: adding, subtracting, and multiplying polynomials, factoring, zeros and roots, and the relationship between factors and zeros — with step-by-step examples, worked problems, and 2 free practice quizzes.
Practice these quizzes
Free · No signupWhat is a polynomial?
A polynomial is an expression built from variables and coefficients using only addition, subtraction, and non-negative whole-number exponents — terms like 3x², −5x, and 7. The degree of a polynomial is the highest exponent present.
| Term count | Name | Example |
|---|---|---|
| 1 term | Monomial | 5x² |
| 2 terms | Binomial | x² + 4 |
| 3 terms | Trinomial | x² + 5x + 6 |
Adding & subtracting polynomials
Subtract: (5x² + 3x − 2) − (2x² − 4x + 7)
When subtracting a polynomial, distribute the negative sign to every term inside the second set of parentheses — not just the first one.
Multiplying polynomials
Multiply: (3x − 2)(x + 5)
For a binomial times a trinomial, distribute every term of the first expression to every term of the second — six total products — then combine like terms.
Factoring polynomials
GCF
Pull out the largest factor shared by every term: 6x³ + 9x² = 3x²(2x + 3)
Trinomial factoring
x² + bx + c factors into (x + m)(x + n), where m · n = c and m + n = b
Difference of squares
a² − b² = (a + b)(a − b)
Factor: x² − 2x − 15
Zeros & roots
A zero (or root) of a polynomial is a value of x that makes the polynomial equal to zero — exactly where its graph crosses the x-axis.
Find the zeros of f(x) = (x − 4)(x + 1)(x − 6).
Factors & zeros: the connection
Every factor of a polynomial corresponds directly to a zero, and vice versa: if (x − k) is a factor, then k is a zero — and if k is a zero, then (x − k) must be a factor.
A polynomial has zeros at x = 2 and x = −5. Write a possible factored form.
A zero of x = −5 corresponds to the factor (x + 5), not (x − 5). Always flip the sign when converting between a zero and its matching factor.
Common PSAT traps
- Sign errors when subtracting polynomials: distribute the negative to every term in the second polynomial.
- Incomplete distribution when multiplying: every term in the first factor must multiply every term in the second.
- Flipping the sign between zeros and factors: a zero of k always pairs with the factor (x − k), regardless of whether k itself is positive or negative.
- Forgetting to check for a GCF before other factoring methods: pulling out a GCF first often simplifies a harder factoring problem into an easy one.
Test day strategy for polynomials
| Question signal | Fastest approach |
|---|---|
| Adding/subtracting polynomials | Combine only like terms; distribute a negative fully before subtracting |
| Multiplying polynomials | Distribute every term of one factor across every term of the other |
| Factoring a trinomial | Check for a GCF first, then find two numbers matching the sum/product pattern |
| "Find the zeros" question | Set the polynomial (or each factor) equal to zero and solve |
| Given zeros, asked for the polynomial | Flip each zero's sign to build the matching factor |
Now put it to work
Two quiz sets, each building on the last — start with Quiz 1 and work through in order, or jump straight to the topic you need.
Polynomial Operations & Zeros 1
Adding, subtracting, multiplying, and factoring polynomials.
Start quiz → Quiz 2Polynomial Operations & Zeros 2
Zeros, roots, and the relationship between factors and zeros.
Start quiz →