How do you use the remainder theorem to evaluate #f(x)=x^5-47x^3-16x^2+8x+52# at x=7?

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Answer 1 · 2017-01-16T14:35:20

#f(7)=10#

Remainder theorem states that if a polynomial #f(x)# is divided by #(x-a)#, the the remainder is #f(a)#.

Hence to evaluate #f(x)=x^5-47x^3-16x^2+8x+52# at #x=7#, we need to divide #f(x)=x^5-47x^3-16x^2+8x+52# by #(x-7)#.

Now, #f(x)=x^5-47x^3-16x^2+8x+52#

#=x^4(x-7)+7x^3(x-7)+2x^2(x-7)-2x(x-7)-6(x-7)+10#

#=(x^4+7x^3+2x^2-2x-6)(x-7)+10#

As such the remainder on dividing #f(x)# by #(x-7)# is #10#

hence #f(7)=10#

Check #f(6)=7^5-47xx7^3-16xx7^2+8xx7+52#

#=16807-16121-784+56+52=10#