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A student uses the division shown to divide -3x^4 + 15x^3 - x + 5 by x - 5, and concludes that x -5 is not a factor of the polynomial. Describe two errors the student made. Is x - 5 a factor? Explain.

A student uses the division shown to divide 3x4 15x3 x 5 by x 5 and concludes that x 5 is not a factor of the polynomial Describe two errors the student made Is class=

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ANSWER

P(3) = 0

Step-by-step explanation:

Factor Theorem is a consequence of Remainder Theorem.

Remainder Theorem states that if polynomial f(x) is divided by a binomial (x - a) then the remainder is f(a).

Factor Theorem states that if f(a) = 0, then the binomial (x - a) is a factor of f(x).

We have the polynomial

P(x) = x^5-3x^4+5x^3-15x^2-6x+18P(x)=x

5

−3x

4

+5x

3

−15x

2

−6x+18

To prove that x-3 is a factor of P, we calculate P(3):

P(3) = 3^5-3*3^4+5*3^3-15*3^2-6*3+18P(3)=3

5

−3∗3

4

+5∗3

3

−15∗3

2

−6∗3+18

P(3) = 243-243+135-135-18+18P(3)=243−243+135−135−18+18

P(3) = 0P(3)=0

Thus, x-3 is a factor of P(x)

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