How do you use the remainder theorem to evaluate f(x)=x^5-47x^3-16x^2+8x+52f(x)=x5−47x3−16x2+8x+52 at x=7?

1 Answer
Shwetank Mauria
Jan 16, 2017

f(7)=10f(7)=10

Explanation:

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

Hence to evaluate f(x)=x^5-47x^3-16x^2+8x+52f(x)=x5−47x3−16x2+8x+52 at x=7x=7, we need to divide f(x)=x^5-47x^3-16x^2+8x+52f(x)=x5−47x3−16x2+8x+52 by (x-7)(x−7).

Now, f(x)=x^5-47x^3-16x^2+8x+52f(x)=x5−47x3−16x2+8x+52

=x^4(x-7)+7x^3(x-7)+2x^2(x-7)-2x(x-7)-6(x-7)+10=x4(x−7)+7x3(x−7)+2x2(x−7)−2x(x−7)−6(x−7)+10

=(x^4+7x^3+2x^2-2x-6)(x-7)+10=(x4+7x3+2x2−2x−6)(x−7)+10

As such the remainder on dividing f(x)f(x) by (x-7)(x−7) is 1010

hence f(7)=10f(7)=10

Check f(6)=7^5-47xx7^3-16xx7^2+8xx7+52f(6)=75−47×73−16×72+8×7+52

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

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