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Finding Equations of Polynomial Functions with Given Zeros: Solving for the Coefficients, Lecture notes of Algebra

How to find the equation of a polynomial function given its zeros using the factored form of a polynomial. It provides examples and practice problems to help understand the concept. The document also discusses the concept of multiplicity and how it affects the polynomial equation.

What you will learn

  • How do you find the equation of a polynomial function given its zeros?
  • What is the concept of multiplicity in polynomial functions and how does it affect the polynomial equation?

Typology: Lecture notes

2021/2022

Uploaded on 09/12/2022

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Finding Equations of Polynomial Functions with Given Zeros
Polynomials are functions of general form 𝑃(𝑥) = 𝑎𝑛𝑥𝑛+ 𝑎𝑛−1 𝑥𝑛−1 + + 𝑎2𝑥2+ 𝑎1𝑥 + 𝑎0
(𝑛 𝑤ℎ𝑜𝑙𝑒 #𝑠)
Polynomials can also be written in factored form 𝑃(𝑥)= 𝑎(𝑥 𝑧1)(𝑥 𝑧2)(𝑥 𝑧𝑖) (𝑎 ℝ)
Given a list of “zeros”, it is possible to find a polynomial function that has these specific zeros. In fact, there
are multiple polynomials that will work. In order to determine an exact polynomial, the “zeros” and a point
on the polynomial must be provided.
Examples: Practice finding polynomial equations in general form with the given zeros.
Find an* equation of a polynomial with the
following two zeros: 𝑥 = −2, 𝑥 = 4
Step 1: Start with the factored form of a
polynomial.
𝑃(𝑥)= 𝑎(𝑥 𝑧1)(𝑥 𝑧2)
Step 2: Insert the given zeros and simplify.
𝑃(𝑥)= 𝑎(𝑥 (−2))(𝑥 4)
𝑃(𝑥)= 𝑎(𝑥 + 2)(𝑥 4)
Step 3: Multiply the factored terms
together.
𝑃(𝑥)= 𝑎(𝑥2 2𝑥 8)
Step 4: The answer can be left with the
generic “𝑎”, or a value for “𝑎can be chosen,
inserted, and distributed.
i.e. if 𝑎 = 1, 𝑡ℎ𝑒𝑛 𝑃(𝑥) = 𝑥2 2𝑥 8
i.e. if 𝑎 = −2, 𝑡ℎ𝑒𝑛 𝑃(𝑥)= −2𝑥2+ 4𝑥 + 16
*Each different choice for 𝑎will result in a
distinct polynomial. Thus, there are an
infinite number of polynomials with the two
zeros 𝑥 = −2 𝑎𝑛𝑑 𝑥 = 4.
Find the equation of a polynomial with the
following zeroes: 𝑥 = 0, 2, 2 that goes
through the point (−2, 1).
Step 1: Start with the factored form of a
polynomial.
𝑃(𝑥)= 𝑎(𝑥 𝑧1)(𝑥 𝑧2)(𝑥 𝑧3)
Step 2: Insert the given zeros and simplify.
𝑃(𝑥)= 𝑎(𝑥 0)(𝑥 (−2))(𝑥 2)
𝑃(𝑥)=𝑎𝑥(𝑥 + 2)(𝑥 2)
Step 3: Multiply the factored terms together
𝑃(𝑥)= 𝑎(𝑥3 2𝑥)
Step 4: Insert the given point (−2, 1) to
solve for “𝑎 .
1 = 𝑎[(−2)3 2(−2)]
1 = 𝑎[−8 + 4]
1 = −4𝑎
𝑎 = 1
4
Step 5: Insert the value for 𝑎 into the
polynomial, distribute, and re-write the
function.
𝑃(𝑥)= 1
4(𝑥3 2𝑥)= 1
4𝑥3+1
2𝑥
Denote the given zeros as 𝑧1 𝑎𝑛𝑑 𝑧2
Denote the given zeros as 𝑧1, 𝑧2𝑎𝑛𝑑 𝑧3
pf3

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Finding Equations of Polynomial Functions with Given Zeros

Polynomials are functions of general form  𝑃(𝑥) = 𝑎

𝑛

𝑥

𝑛

  • 𝑎

𝑛− 1

𝑥

𝑛− 1

  • ⋯ + 𝑎

2

𝑥

2

  • 𝑎

1

𝑥 + 𝑎

0

(𝑛 ∈ 𝑤ℎ𝑜𝑙𝑒 #

𝑠)

Polynomials can also be written in factored form  𝑃(𝑥) = 𝑎(𝑥 − 𝑧 1

)(𝑥 − 𝑧

2

) … (𝑥 − 𝑧

𝑖

) (𝑎 ∈ ℝ)

Given a list of “zeros”, it is possible to find a polynomial function that has these specific zeros. In fact, there

are multiple polynomials that will work. In order to determine an exact polynomial , the “zeros” and a point

on the polynomial must be provided.

Examples: Practice finding polynomial equations in general form with the given zeros.

Find an equation of a polynomial* with the

following two zeros: 𝑥 = − 2 , 𝑥 = 4

Step 1: Start with the factored form of a

polynomial.

𝑃(𝑥) = 𝑎(𝑥 − 𝑧

1

)(𝑥 − 𝑧

2

)

Step 2: Insert the given zeros and simplify.

𝑃(𝑥) = 𝑎(𝑥 − (− 2 ))(𝑥 − 4 )

𝑃

( 𝑥

) = 𝑎(𝑥 + 2 )(𝑥 − 4 )

Step 3: Multiply the factored terms

together.

𝑃

( 𝑥

) = 𝑎(𝑥

2

− 2 𝑥 − 8 )

Step 4: The answer can be left with the

generic “𝑎”, or a value for “𝑎”can be chosen,

inserted, and distributed.

i.e. if 𝑎 = 1 , 𝑡ℎ𝑒𝑛 𝑃(𝑥) = 𝑥

2

− 2 𝑥 − 8

i.e. if 𝑎 = − 2 , 𝑡ℎ𝑒𝑛 𝑃(𝑥) = − 2 𝑥

2

  • 4 𝑥 + 16

*Each different choice for “ 𝑎 ” will result in a

distinct polynomial. Thus, there are an

infinite number of polynomials with the two

zeros 𝑥 = − 2 𝑎𝑛𝑑 𝑥 = 4_._

Find the equation of a polynomial with the

following zeroes: 𝑥 = 0 , − √

2 that goes

through the point (− 2 , 1 ).

Step 1: Start with the factored form of a

polynomial.

𝑃

( 𝑥

) = 𝑎

( 𝑥 − 𝑧

1

)( 𝑥 − 𝑧

2

)( 𝑥 − 𝑧

3

)

Step 2: Insert the given zeros and simplify.

𝑃

( 𝑥

) = 𝑎(𝑥 − 0 )(𝑥 − (−√ 2 ))(𝑥 − √ 2 )

𝑃(𝑥) = 𝑎𝑥(𝑥 + √

2 )(𝑥 − √

2 )

Step 3: Multiply the factored terms together

𝑃

( 𝑥

) = 𝑎(𝑥

3

− 2 𝑥)

Step 4: Insert the given point (− 2 , 1 ) to

solve for “𝑎 “.

1 = 𝑎[(− 2 )

3

− 2 (− 2 )]

1 = 𝑎

[ − 8 + 4

]

1 = − 4 𝑎

𝑎 = −

1

4

Step 5: Insert the value for 𝑎 ” into the

polynomial, distribute , and re-write the

function.

𝑃

( 𝑥

) = −

1

4

( 𝑥

3

− 2 𝑥

) = −

1

4

𝑥

3

1

2

𝑥

Denote the given zeros as 𝑧

1

𝑎𝑛𝑑 𝑧

2

Denote the given zeros as 𝑧

1

, 𝑧

2

𝑎𝑛𝑑 𝑧

3

Polynomials can have zeros with multiplicities greater than 1. This is easier to see if the

Polynomial is written in factored form.

1

𝑚

2

𝑛

𝑖

𝑝

Multiplicity - The number of times a “zero” is repeated in a polynomial. The multiplicity of

each zero is inserted as an exponent of the factor associated with the zero. If the multiplicity

is not given for a zero, it is assumed to be 1.

Examples: Practice finding polynomial equations with the given zeros and multiplicities.

Find an equation of a polynomial with the

given zeroes and associated multiplicities.

Leave the answer in factored form.

Zeros Multiplicity

𝑥 = 1 2

𝑥 = − 2 3

𝑥 = 3 1

Step 1: Write the factored form of the

Polynomial.

𝑃(𝑥) = 𝑎(𝑥 − 𝑧

1

)

𝑚

(𝑥 − 𝑧

2

)

𝑛

(𝑥 − 𝑧

3

)

𝑝

Step 2: Insert the given zeros and their

corresponding multiplicities.

𝑃

( 𝑥

) = 𝑎

( 𝑥 − 1

)

2

( 𝑥 − − 2

)

3

( 𝑥 − 3

)

1

Step 3: Simplify any algebra if necessary. The

answer can be left with the generic “𝑎”, or a

specific value for “𝑎”can be chosen and

inserted if requested.

𝑃

( 𝑥

) = 𝑎

( 𝑥 − 1

)

2

( 𝑥 + 2

)

3

(𝑥 − 3 )

i.e. let 𝑎 = 1 , 𝑡ℎ𝑒𝑛

𝑃(𝑥) = (𝑥 − 1 )

2

(𝑥 + 2 )

3

(𝑥 − 3 )

let 𝑎 = − 2 , 𝑡ℎ𝑒𝑛

𝑃(𝑥) = − 2 (𝑥 − 1 )

2

(𝑥 + 2 )

3

(𝑥 − 3 )

Find an equation of a polynomial with the

given zeros and associated multiplicities.

Expand the answer into general form.

Zeros Multiplicity

𝑥 = 0 3

𝑥 = − 1 2

𝑥 = 𝑖 1

𝑥 = −𝑖 1

Step 1: Write the factored form of the

Polynomial.

𝑃

( 𝑥

) = 𝑎

( 𝑥 − 𝑧

1

)

𝑚

( 𝑥 − 𝑧

2

)

𝑛

( 𝑥 − 𝑧

𝑖

)

𝑝

Step 2: Insert the given zeros and their

corresponding multiplicities.

𝑃

( 𝑥

) = 𝑎

( 𝑥 − 0

)

3

( 𝑥 − − 1

)

2

( 𝑥 − 𝑖

)

1

( 𝑥 − −𝑖

)

1

Step 3: Simplify any algebra if necessary.

𝑃

( 𝑥

) = 𝑎𝑥

3

( 𝑥 + 1

)

2

(𝑥 − 𝑖)(𝑥 + 𝑖)

Step 4 : Multiply the factored terms

together. Recall that 𝑖

2

= − 1! Note the

generic “𝑎” can be used and distributed, or a

specific value for “𝑎” can be chosen and

inserted if requested.

𝑃(𝑥) = 𝑎𝑥

7

  • 2 𝑎𝑥

6

  • 2 𝑎𝑥

5

  • 2 𝑎𝑥

4

  • 𝑎𝑥

3

Insert Multiplicities of each zero

Insert zeros

Zeros can

be real or

imaginary