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Chapter 4: Q. 18 (page 209)

Use the Remainder Theorem to find the remainder whenf(x)is divided by x-c. Then use the Factor Theorem to determine whether x-cis a factor of f(x).

f(x)=x6-16x4+x2-16;x+4

Short Answer

Expert verified

The remainder is 0and x+4is a factor.

Step by step solution

01

Step 1. Given information.

The given function is,

f(x)=x6-16x4+x2-16;x+4

02

Step 2. Using remainder theorem.

We know that if f(x)is divided by x-cthen the remainder will be f(c).

Here f(x)is divided by x+4, then the remainder is f(-4)that is

f(-4)=-46-16-44+-42-16=4096-16256+16-16=4096-4096=0

Therefore, the remainder is 0.

03

Step 3. Using factor theorem.

We know that if x-cis the factor of the polynomial f(x)then

f(c)=0,

Here c=-4, then

f(c)=f(-4)=0

We observe that f(-4)=0,then x+4is a factor.

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