Factor the polynomial completely, and find all its zeros. State

Stacie Worsley

Stacie Worsley

Answered question

2021-12-07

Factor the polynomial completely, and find all its zeros. State the multiplicity of each zero.
P(x)=16x481

Answer & Explanation

Neunassauk8

Neunassauk8

Beginner2021-12-08Added 30 answers

Step 1 
Given polynomial is P(x)=16x481
To fully factor the polynomial, locate each of its zeros, and establish multiplicity.
Solution: 
We have identity, 
a2b2=(a+b)(ab) 
Factorizing the given polynomial. 
P(x)=16x481 
=(4x2)2(9)2 
=(4x2+9)(4x29) 
=(4x2+9)((2x)232) 
=(4x2+9)(2x+3)(2x3) 
Factorization of the given polynomial is P(x)=(4x2+9)(2x+3)(2x3)
Step 2 
Now locate the polynomial's zeros.
(4x2+9)(2x+3)(2x3)=0 
Zero product rule says that if a·b=0 then either a=0 or b=0. 
So, we will have: 
(4x2+9)=0 or (2x+3)=0 or (2x-3)=0 
4x2=9 or 2x=-3 or 2x=3 
x=94  or  x=32  or  x=32 
x=±94 
x=±3i2 
Therefore, zeroes of the polynomial are 32i,32i,32  and  32
Step 3 
The number of zero repetitions in a factor is known as the multiplicity of zeroes.
Here, we can find that all the zeroes occurs for only time. 
Hence, multiplicity of all the zeroes is one.

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