Theorems.........Please help?

Question

Use the remainder theorem and the factor theorem to determine whether (b+4) is a factor of (b^3+3b^-b+12)

A. The remainder is 0 and, therefore, (b+4) is a factor of (b^3+3b^-b+12)

B. The remainder is 0 and, therefore, (b+4) isn't a factor of (b^3+3b^-b+12)

C. The remainder isn't 0 and, therefore, (b+4) is a factor of (b^3+3b^-b+12)

D. The remainder isn't 0 and, therefore, (b+4) isn't a factor of (b^3+3b^-b+12)



2. Use the remainder theorem and the factor theorem to determine whether (c+5) is a factor of (c^4+7c^3+6c^2-18c+10)

A. The remainder is 0 and, therefore, (c+5) is a factor of (c^4+7c^3+6c^2-18c+10)

B. The remainder is 0 and, therefore, (c+5) isn't a factor of (c^4+7c^3+6c^2-18c+10)

C. The remainder isn't 0 and, therefore, (c+5) is a factor of (c^4+7c^3+6c^2-18c+10)

D. The remainder isn't 0 and, therefore, (c+5) isn't a factor of (c^4+7c^3+6c^2-18c+10)


THANKS

Answer 1

X is a factor of Y if and only if there's a Z such that XZ = Y.

I.e., it's a factor if and only if the remainder = 0.

As for checking whether or not the remainder truly IS zero, do polynomial long division. For example, in problem 2 the first term is c^3. Then subtract (c+5)*1*c^3 from the quartic polynomial to get a cubic one whose first term is 2*c^3. So the second term of the quotient is 2*c^2. Keep going.

Answer 2

If x is a factor of y, then y/x gives a remainder of zero. Therefore, the B and C answers in all these questions make no sense. So it's a matter of finding out whether in each case the remainder is zero, then choosing A or D based off of that.

The remainder theorem says that if you divide a polynomial by (x-a), then the reaminder is equal to the answer you get when you plug x=a into the polynomial. In the first example, you have (b^3+3b^-b+12) as the polynomial, but I'll assume that's a typo and you mean (b^3 + 3b^2 - b + 12). So to see if (b+4) is a factor, plug b=-4 into the equation to get -64 + 48 + 4 + 12 = 0. This means the remainder is zero, and thus b+4 is a factor of the polynomial.

Apply the same rule to the other questions to get the answers.

Answer 3

let f(b)=(b^3+3b^2-b+12)
if (b+4=>b=-4) is factor of f(b) then f(b)=0=> f(-4)=0
here f(-4)=-64+48+4+12=>f(-4)=0
therefore b=-4 is a factor of f(b)


similarily the below one.