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Equivalent Representations of Polynomial Expressions (College Board AP® Precalculus): Study Guide

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 9 April 2026

Equivalent representations of polynomial expressions

What are the different forms of a polynomial expression?

A polynomial function can be written in different equivalent forms
- each of which reveals different information about the function
The two most important forms for polynomial functions are standard form and factored form

What is standard form?

The standard form of a polynomial is the expanded form written as a sum of terms in descending order of degree
- $p (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + a_{n - 2} x^{n - 2} + \dots + a_{2} x^{2} + a_{1} x + a_{0}$
  - where $a_{n} \neq 0$
- This is the same form introduced in the Polynomial Functions study guide
Standard form makes it easy to see
- The degree of the polynomial (the highest power of $x$ )
- The leading term ( $a_{n} x^{n}$ ) and the leading coefficient ( $a_{n}$ )
- The end behavior of the function (determined by the degree and sign of the leading coefficient, as covered in the End Behavior of Polynomial Functions study guide)
- The constant term ( $a_{0}$ ), which gives the $y$ -intercept of the graph (since $p (0) = a_{0}$ )
  - so the $y$ -intercept is at $(0, a_{0})$

What is factored form?

The factored form of a polynomial expresses the function as a product of linear factors
- E.g. $p (x) = - 2 (x - 1) (x + 3) (x - 5)$
- There may also be an irreducible quadratic factor if there are non-real zeros
  - E.g. $p (x) = 3 (x - 2) (x^{2} + 2 x + 5)$
Factored form makes it easy to see
- The real zeros of the function (the values of $x$ that make each factor equal to zero)
- The multiplicity of each zero (from repeated factors)
- The $x$ -intercepts of the graph (since the real zeros correspond to points where the graph crosses or touches the $x$ -axis)
- E.g. from $p (x) = - 2 (x - 1) (x + 3) (x - 5)$
  - The zeros are $x = 1$ , $x = - 3$ , and $x = 5$ (each with multiplicity 1)
  - The $x$ -intercepts are $(1, 0)$ , $(- 3, 0)$ , and $(5, 0)$

How do I convert between forms?

From factored form to standard form
- Expand (multiply out) the factors and collect like terms
- E.g. $p (x) = (x - 2) (x + 5) = x^{2} + 5 x - 2 x - 10 = x^{2} + 3 x - 10$
From standard form to factored form
- Factor the polynomial expression
- For this course, factoring without technology is limited to:
  - Factoring out a common factor
    - E.g. $3 x^{3} - 6 x^{2} = 3 x^{2} (x - 2)$
  - Factoring quadratics
    - E.g. $x^{2} + 3 x - 10 = (x + 5) (x - 2)$
- For higher-degree polynomials, you may need to combine these techniques
  - E.g. $p (x) = x^{3} + 2 x^{2} - 15 x = x (x^{2} + 2 x - 15) = x (x + 5) (x - 3)$
  - First factor out the common factor $x$ , then factor the remaining quadratic

Examiner Tips and Tricks

On a calculator part of the exam, a graphing calculator can be used to identify zeros. These can then be used to write the factored form.

Why is it useful to have different forms?

Different forms make different information accessible
For example
- If you need to determine end behavior, use standard form
  - the degree and leading coefficient are immediately visible
- If you need to find zeros or $x$ -intercepts, use factored form
  - the zeros can be read directly from the factors
- If you need to find the $y$ -intercept, either form works
  - but standard form makes it easiest (the constant term $a_{0}$ )
Being able to convert between forms means you can extract whatever information is needed from a single polynomial function

Examiner Tips and Tricks

When a question gives you a polynomial in one form but asks about a property that is more easily seen in the other form, convert first.

For example, if you are given a polynomial in factored form and asked about end behavior, multiply out the leading terms to identify the degree and leading coefficient
You don't need to fully expand the entire expression here!
E.g. for $p (x) = - 2 (x - 1) (x + 3) (x - 5)$ , the leading term is $- 2 \cdot x \cdot x \cdot x = - 2 x^{3}$
- so the degree is 3 and the leading coefficient is $- 2$ .

Worked Example

The polynomial function $p$ is given by $p (x) = (x + 4) (x^{2} - 2 x - 8)$ .

(a) Rewrite $p (x)$ in factored form as a product of three linear factors.

Answer:

Factor the quadratic $x^{2} - 2 x - 8$

Find two numbers that multiply to $- 8$ and add to $- 2$
- these are $- 4$ and $2$ .

$x^{2} - 2 x - 8 = (x - 4) (x + 2)$

Substitute back into the original expression

$p (x) = (x + 4) (x - 4) (x + 2)$

(b) Find all zeros of $p$ and identify the $x$ -intercepts of the graph of $p$ .

Answer:

Set each factor in the factored form equal to zero

$x + 4 = 0 \Rightarrow x = - 4$

$x - 4 = 0 \Rightarrow x = 4$

$x + 2 = 0 \Rightarrow x = - 2$

The zeros of $p$ are $x = - 4$ , $x = - 2$ , and $x = 4$

And therefore

The $x$ -intercepts are $(- 4, 0)$ , $(- 2, 0)$ , and $(4, 0)$

Answer:

To determine end behavior, identify the degree and leading coefficient

From the factored form, the leading term is the product of the leading terms of each factor

$x \cdot x \cdot x = x^{3}$

So $p$ has degree $3$ (odd)

with a positive leading coefficient ( $1$ )

$\lim_{x \to - \infty} p (x) = - \infty$ and $\lim_{x \to \infty} p (x) = \infty$

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Equivalent Representations of Polynomial Expressions (College Board AP® Precalculus): Study Guide

Equivalent representations of polynomial expressions

What are the different forms of a polynomial expression?

What is standard form?

What is factored form?

How do I convert between forms?

Examiner Tips and Tricks

Why is it useful to have different forms?

Examiner Tips and Tricks

Worked Example

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Author: Roger B

Reviewer: Mark Curtis