Answer:
[tex]k=1\text{ or } k=4[/tex]
Step-by-step 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:
[tex](x^2+5x+7)\div(x+k)[/tex]
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:
[tex](-k)^2+5(-k)+7=3[/tex]
Simplify:
[tex]k^2-5k+7=3[/tex]
Solve for k. Subtract 3 from both sides:
[tex]k^2-5k+4=0[/tex]
Factor:
[tex](k-1)(k-4)=0[/tex]
Zero Product Property:
[tex]k-1=0\text{ or } k-4=0[/tex]
Solve:
[tex]k=1\text{ or } k=4[/tex]
So, either of the two expressions:
[tex](x^2+5x+7)\div(x+1)\text{ or } (x^2+5x+7)\div(x+4)[/tex]
Will yield 3 as the remainder.
