AlgebraAlgebra 1High school

Polynomials and Factoring

Add, multiply and factor polynomials, including GCF, grouping and trinomials.

A polynomial is a sum of terms of the form $a x^n$ with whole-number exponents, such as $4x^3 - 2x + 7$ . The degree is the highest exponent, and it controls the shape of the graph: the end behaviour, the maximum number of turning points and the maximum number of real roots.

The core skills are adding and subtracting (combine like terms), multiplying (distribute every term over every term) and factoring (reverse the multiplication). Factoring is the key to solving polynomial equations, because a product equals zero only when one factor does.

Key terms

Difference of squares: A difference of two squares factors into conjugates.
$a^2 - b^2 = (a-b)(a+b)$

Key ideas

Always factor out the greatest common factor first.
Recognise special products: $a^2 - b^2 = (a - b)(a + b)$ and $a^2 \pm 2ab + b^2 = (a \pm b)^2$ .
The remainder theorem: dividing $p(x)$ by $x - r$ leaves remainder $p(r)$ .
A polynomial of degree $n$ has at most $n$ real roots.

Worked example

Factor completely $2x^3 - 8x$ .

Common factor: $2x(x^2 - 4)$ .
Difference of squares: $x^2 - 4 = (x - 2)(x + 2)$ .
Result: $2x(x - 2)(x + 2)$ .
Check by expanding: $2x(x^2 - 4) = 2x^3 - 8x$ .

Answer: $2x(x - 2)(x + 2)$

Common mistakes

Combining unlike terms such as $x^2$ and $x$ .
Forgetting the middle term when squaring: $(x + 3)^2 \ne x^2 + 9$ .
Stopping after the GCF when the remaining factor still factors.

Practice questions

Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.

1.What is the coefficient of $x$ in $(x + 4)(x - 7)$ ?
Show answer
$-3$
2.What is the remainder when $x^3 - 2x + 5$ is divided by $x - 2$ ?
Show answer
$9$
3.What is the degree of $(x^2 + 1)(x^3 - x)$ ?
Show answer
$5$

Polynomials and Factoring

Key terms

Key ideas

Worked example

Common mistakes

Practice questions

Keep practising

Calculators

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