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Find the values of k so that each remainder is three.
10. (x^2+ 5x + 7) = (x + k)

Find the values of k so that each remainder is three. 10. (x^2+ 5x + 7) = (x + k)-example-1

1 Answer

11 votes

Answer:


k=1\text{ or } k=4

Explanation:

We can use the Polynomial Remainder Theorem. It states that if we divide a polynomial P(x) by a binomial in the form (x - a), then our remainder will be P(a).

We are dividing:


(x^2+5x+7)/(x+k)

So, a polynomial by a binomial factor.

Our factor is (x + k) or (x - (-k)). Using the form (x - a), our a = -k.

We want our remainder to be 3. So, P(a)=P(-k)=3.

Therefore:


(-k)^2+5(-k)+7=3

Simplify:


k^2-5k+7=3

Solve for k. Subtract 3 from both sides:


k^2-5k+4=0

Factor:


(k-1)(k-4)=0

Zero Product Property:


k-1=0\text{ or } k-4=0

Solve:


k=1\text{ or } k=4

So, either of the two expressions:


(x^2+5x+7)/(x+1)\text{ or } (x^2+5x+7)/(x+4)

Will yield 3 as the remainder.

User Hsop
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