I need help understanding how to process to polynomial questions in regards to two unknown values. 1.) When 3x^3 + mx^2 + nx +2 is divided by (x + 2), the remainder is 8. When the same polynomial is divided by (x - 1), the remainder is 2. Determine the values of m and n. 2.) When 2x^3 + mx^2 + nx - 6 is divided by (x - 2), the remainder is 20. The same polynomial has a factor of (x + 2). Determine the values of m and n. I'm new to this site and I accidently Brained somebody and didn't get my answer.
Answers
Step-by-step explanation:
ANSWER
Let f(x)=2x
3
−kx
2
+7x−1.
Now according to the remainder theorem, when f(x) is divided by x−1 then the remainder is f(1).
Now, f(1)=2−k+7−1=8−k.
According to the problem,
8−k=3
or, k=5.
Step-by-step explanation:
1. If x−1 is a factor that means that when x=−1, the function goes to zero. Hence,
6(−1)3+m(−1)2+n(−1)−5=0⇒m−n=11
If we divide by x−1 and have a remainder of 4, that means that the value of the expression at x=1 is 4. Therefore,
6(1)3+m(1)2+n(1)−5=4⇒m+n=3
We have two equations and two unknowns, hence we can solve for m and n. We add then and divide by two to find that m=7 and n=−4.
SIDENOTE:
If we want to prove the second piece of logic (that x=1 makes the expression 4), we can imagine the expression minus 4 to now have a factor of x−1 (by definition of remainder) and we can use the logic from above. Hope that makes sense!