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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
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Polynomials are functions of general form 𝑃(𝑥) = 𝑎
𝑛
𝑥
𝑛
𝑛− 1
𝑥
𝑛− 1
2
𝑥
2
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
*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
6
5
4
3
Insert Multiplicities of each zero
Insert zeros
Zeros can
be real or
imaginary