Build polynomial equations from given conditions

Ontario Grade 12 Mathematics

Polynomial and Rational Functions

Turn information about roots, points, and graph features into a polynomial model.

A polynomial equation can describe a graph or a set of values. In this lesson, you will build a polynomial equation from information such as its zeros and a point on its graph. First, recall that a zero is an input value that makes a function equal to zero. For example, if

f(3)=0

, then

3

is a zero of

f

. A polynomial may also be written in factored form, which makes its zeros visible. The main idea is to translate each condition into part of an equation, then use all the conditions to complete the model.

What you will learn

Recognize polynomial conditions that can be written as equations.
Build a polynomial using known zeros and their multiplicities.
Use a point on the graph to determine an unknown coefficient.
Check that the resulting polynomial satisfies every given condition.

1. Prerequisite bridge: factors, zeros, and degree

A polynomial is an expression made from constants and variables raised to whole-number powers, combined using addition, subtraction, and multiplication. For example,

2x^3-x+4

is a polynomial.

A factor is an expression that is multiplied by another expression. If a polynomial has a factor

x-2

, then substituting

x=2

makes that factor equal to zero. The whole product is therefore zero, so

2

is a zero of the polynomial.

The degree of a polynomial is the greatest exponent of the variable after the expression is simplified. For a polynomial written as a product of linear factors, the degree is the number of those factors, counting repeats. A repeated factor indicates a repeated zero. For example,

(x+1)^2

has the zero

-1

repeated twice.

x-r=0\iff x=r

The factor $x-r$ gives the zero $r$ .
A repeated factor represents a zero with multiplicity greater than one.
The degree can help decide how many factors a model needs.

2. Translate conditions into factors

A condition is information the polynomial must satisfy. A stated zero gives a factor directly: if

r

is a zero, include

x-r

. If a zero is repeated, include the corresponding factor more than once.

Suppose a polynomial has zeros

-2

and

4

, and the zero

-2

is repeated. The known factors are

(x+2)^2

and

(x-4)

. Their product is a starting model. If the polynomial is known to have degree

3

, these factors already account for the full degree.

Sometimes the polynomial has a larger degree than the factors supplied by the zeros. In that case, more information is needed to determine the remaining coefficient or factor. A point on the graph is useful because its coordinates tell you an input and the corresponding output.

f(x)=a(x-r_1)(x-r_2)\cdots(x-r_n)

A zero $r$ contributes the factor $x-r$ .
A repeated zero contributes a repeated factor.
Use the stated degree to check whether the factors account for all powers.

3. Use a point to determine the scale

The coefficient

a

in a factored model is often unknown. It changes the vertical scale of the graph without changing the listed zeros. To find it, substitute the coordinates of a known point into the model.

A point

(p,q)

on the graph means that the function value at input

p

is

q

. Substitute

x=p

and set the expression equal to

q

. Solve the resulting equation for the unknown coefficient.

This approach works because the point condition and the zero conditions describe the same polynomial. The final equation should satisfy both kinds of information. Check by substituting the zeros and the given point into the completed model.

f(p)=q

A point $(p,q)$ means $f(p)=q$ .
Substitution turns the point condition into an equation for an unknown coefficient.
Check every given condition in the completed polynomial.

Condition-to-model guide

Given condition	How it enters the model	Example
Zero $r$	Include the factor $x-r$	Zero $5$ gives $x-5$
Repeated zero $r$	Repeat its factor	Zero $-2$ twice gives $(x+2)^2$
Point $(p,q)$	Substitute $x=p$ and set $f(p)=q$	Point $(2,7)$ gives $f(2)=7$
Degree $n$	Count factors, including repeats	Three linear factors give degree $3$

Worked example

Build a cubic from zeros and a point

Build a polynomial equation of degree

3

with zeros

-1

and

3

, where

-1

is repeated, and whose graph passes through

(1,16)

.

Translate the zeros
The zero $-1$ gives the factor $x+1$ . Since it is repeated, use that factor twice. The zero $3$ gives the factor $x-3$ . Together these factors have degree $3$ , as required.
$f(x)=a(x+1)^2(x-3)$
Use the point condition
The graph passes through $(1,16)$ , so the output must be $16$ when the input is $1$ . Substitute those values into the factored model.
$16=a(1+1)^2(1-3)$
Solve for the coefficient
Evaluate the factors. Their product is $-8$ , so the equation becomes $16=-8a$ . Dividing both sides by $-8$ gives the coefficient.
$a=-2$
Write the polynomial equation
Replace the unknown coefficient with $-2$ . The factored form clearly displays the zeros and their repetition.
$f(x)=-2(x+1)^2(x-3)$

Answer: The polynomial is

f(x)=-2(x+1)^2(x-3)

.

Check: At

x=-1

, the repeated factor is zero; at

x=3

, the other factor is zero. At

x=1

, the value is

-2(2)^2(-2)=16

. The polynomial has degree

3

.

Common mistakes and how to avoid them

Using

x+r

for a zero

r

.

Correction: Use

x-r

. For example, zero

-4

gives

x+4

, since substituting

-4

makes that factor zero.

Forgetting to repeat a factor when a zero is repeated.

Correction: Include the factor the stated number of times. A zero repeated twice gives a squared factor.

Assuming the zeros determine the whole polynomial.

Correction: Zeros determine factors, but an unknown coefficient may remain. Use an additional condition, such as a point, to determine it.

Using the point coordinates in the wrong order.

Correction: For a point

(p,q)

, substitute

p

for the input and set the result equal to

q

.

Lesson summary

Translate each zero $r$ into a factor $x-r$ .
Repeat factors when the zero is repeated.
Use the stated degree to check the number of factors.
Use a point $(p,q)$ through the condition $f(p)=q$ to find an unknown coefficient.
Check the final polynomial against all given conditions.

Check your understanding

Question 1

A polynomial has degree

3

, zeros

2

and

-1

, and the zero

2

is repeated. Which factored model includes these conditions?

$a(x-2)^2(x+1)$
$a(x+2)^2(x-1)$
$a(x-2)(x+1)$
$a(x-2)^2(x-1)$

Show answer and explanation

a(x-2)^2(x+1)

The zero

2

gives

x-2

and is repeated, while zero

-1

gives

x+1

. There are three linear factors, matching degree

3

.

Question 2

A polynomial is given by

f(x)=a(x-2)(x+1)

and passes through

(0,6)

. What is

a

?

$-3$
$-1$
$1$
$3$

Show answer and explanation

-3

Substitute the point condition:

6=a(0-2)(0+1)=-2a

. Dividing by

-2

gives

a=-3

.

Key terms

Polynomial: An expression formed from constants and variables with whole-number exponents, using addition, subtraction, and multiplication.
Zero: An input value that makes a function's output equal to zero.
Factor: An expression that is multiplied by another expression.
Degree: The greatest exponent of the variable in a polynomial after simplification.
Multiplicity: The number of times a zero's factor appears in the polynomial.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation C1.7. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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