Math

Polynomials

6 min readMedium5-question drill

Polynomials are the building blocks of algebra — expressions with terms involving powers of variables. Master the four operations (add, subtract, multiply, factor) and you've unlocked half the SAT Math section.

Where this lesson fits

Before

Linear Equations→

Polynomials

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→Quadratic Equations →Exponents and Radicals

A polynomial is a sum of terms, each of which is a number times a power of a variable: $3x^4 + 2x^2 - 7x + 5$ . The degree is the highest power: this one is degree 4.

Special names by degree:

Degree 0: constant ( $5$ )
Degree 1: linear ( $2x + 1$ )
Degree 2: quadratic ( $x^2 - 4$ )
Degree 3: cubic ( $x^3 + x$ )

Adding and subtracting: combine like terms (terms with the same variable raised to the same power).

$(3x^2 + 2x) + (x^2 - 5x) = 4x^2 - 3x$

When subtracting, distribute the negative to every term: $(3x^2 + 2x) - (x^2 - 5x) = 3x^2 + 2x - x^2 + 5x = 2x^2 + 7x$ .

Multiplying: distribute every term in the first by every term in the second.

$(x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6$

Factoring is multiplying in reverse — turning a polynomial into a product. Three patterns to memorize:

The three factoring patterns to memorize

Pattern	Form	Example
Difference of squares	$a^2 - b^2 = (a+b)(a-b)$	$x^2 - 25 = (x+5)(x-5)$
Perfect square (sum)	$a^2 + 2ab + b^2 = (a+b)^2$	$x^2 + 6x + 9 = (x+3)^2$
Perfect square (diff)	$a^2 - 2ab + b^2 = (a-b)^2$	$x^2 - 6x + 9 = (x-3)^2$
Standard quadratic	$x^2 + bx + c$ → find $p, q$ where $pq = c$, $p + q = b$	$x^2 + 5x + 6 = (x+2)(x+3)$

Pattern 1: Difference of squares. $a^2 - b^2 = (a + b)(a - b)$ .

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

Pattern 2: Perfect square trinomials. $a^2 + 2ab + b^2 = (a + b)^2$ and $a^2 - 2ab + b^2 = (a - b)^2$ .

Example: $x^2 + 6x + 9 = (x + 3)^2$ .

Pattern 3: Standard quadratic factoring. $x^2 + bx + c$ — find two numbers that multiply to $c$ and add to $b$ .

Example: $x^2 + 5x + 6$ . Two numbers that multiply to 6, add to 5: 2 and 3. So $(x + 2)(x + 3)$ .

How should I factor this polynomial?

Is there a GCF (number or variable common to all terms)?

Yes ↓

Pull out the GCF first. Then is what's left $a^2 - b^2$?

Yes ↓

→ Difference of squares: $(a+b)(a-b)$

No ↓

→ Try perfect-square or standard-quadratic factoring on the remaining polynomial

No ↓

Is it $a^2 - b^2$ form (two squares minus)?

Yes ↓

→ Difference of squares: $(a+b)(a-b)$

No ↓

→ Standard quadratic: find two numbers that multiply to $c$, add to $b$

Polynomial division. When dividing $p(x)$ by $(x - a)$ , you can either do long division or — better — use the factor theorem: if $(x - a)$ is a factor, then $p(a) = 0$ .

Example: Is $(x - 2)$ a factor of $x^3 - 4x^2 + x + 6$ ? Plug in $x = 2$ : $8 - 16 + 2 + 6 = 0$ . Yes!

Common SAT setup: "For what values of $k$ does the polynomial have a factor of $(x - 3)$ ?" → plug 3 in for $x$ , solve for $k$ .

Quick check

Identify the factoring pattern (GCF, difference of squares, perfect square, or standard quadratic), then apply it. Always check by FOILing back.

Which expression is equivalent to x² - 16?

(x + 4)(x - 4)

(x + 4)²

(x - 4)²

(x + 16)

Worked examples

Example 1

Factor: $x^2 - 7x + 12$ .

Example 2

If $(x + 4)$ is a factor of $f(x) = x^3 + 2x^2 + kx - 24$ , what is the value of $k$ ?

Common pitfalls

Forgetting to distribute the negative on subtraction

$(3x^2 + 2x) - (x^2 - 5x) = 3x^2 + 2x - x^2 + 5x$ , NOT $3x^2 + 2x - x^2 - 5x$ . Distribute the minus to every term inside the parentheses. This is the #1 sign error on polynomial subtraction.

Confusing $(x + 3)^2$ with $x^2 + 9$

$(x + 3)^2 = x^2 + 6x + 9$ , NOT $x^2 + 9$ . Always FOIL out perfect squares — don't just square the terms separately. Same trap with $(a - b)^2 = a^2 - 2ab + b^2$ , not $a^2 - b^2$ .

Missing the GCF before factoring

$2x^2 + 6x + 4$ : factor out 2 first → $2(x^2 + 3x + 2) = 2(x+1)(x+2)$ . Trying to factor without pulling out the GCF makes the problem harder than it needs to be.

Stopping after one factor

$x^4 - 16$ → first factor as difference of squares: $(x^2 - 4)(x^2 + 4)$ . But $x^2 - 4$ is also a difference of squares: $(x - 2)(x + 2)$ . So fully factored: $(x - 2)(x + 2)(x^2 + 4)$ . Always check if any factor can be factored further.

Key takeaways

A polynomial is a sum of terms with non-negative integer powers of variables. Degree = highest power.
When subtracting polynomials, distribute the negative to every term inside the parentheses.
Three factoring patterns: difference of squares ( $a^2 - b^2$ ), perfect square trinomial ( $a^2 \pm 2ab + b^2$ ), standard quadratic ( $x^2 + bx + c$ ).
Factor theorem: $(x - a)$ is a factor of $p(x)$ if and only if $p(a) = 0$ .
Always pull out the GCF first, and always check whether a factor can be factored further.

Watch & learn

Curated Khan Academy walkthroughs on Polynomials. They're complementary to this lesson — watch one if a written explanation isn't clicking, or after to reinforce.

Try it yourself

5 practice questions on Polynomials, drawn from the question bank. The tutor is one click away if you get stuck.

Where this lesson fits

Before

Linear Equations→

Polynomials

You are here

→Quadratic Equations →Exponents and Radicals