SOLUTION: a) Use the remainder theorem to find the remainder when 𝑓(𝑥)=−4𝑥^3 +5𝑥^2 + 8 is divided by 𝑥+3. Then use the Factor Theorem to determine whether 𝑥 + 3 is

Question 1189262: a) Use the remainder theorem to find the remainder when 𝑓(𝑥)=−4𝑥^3 +5𝑥^2 + 8 is
divided by 𝑥+3. Then use the Factor Theorem to determine whether 𝑥 + 3 is a
factor of 𝑓(𝑥).
b) Use the Rational Zeros Theorem to find all the real zeros of the polynomial
function 𝑓(𝑥)=x^4 -x^3 -6x^2 +4x +8
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

Part (a)

Remainder theorem: Dividing p(x) over (x-k) yields some quotient q(x) and a remainder p(k).
Special case: if the remainder is 0, then (x-k) is a factor of p(x).

Compare (x+3) with (x-k) to see that k = -3
It might help to rewrite x+3 as x-(-3).

f(x) = -4x^3+5x^2+8
f(-3) = -4(-3)^3+5(-3)^2+8
f(-3) = 161

Answers:
The remainder is 161
(x+3) is not a factor of f(x) because we didn't get a remainder of 0.

================================================
Part (b)

Due to the leading coefficient being 1, this means we simply need to list the plus/minus version of each factor of the last term 8

The list of all possible rational roots:
1, -1, 2, -2, 4, -4, 8, -8

Then check each value to see if we get f(x) = 0 or not.

If we tried x = 1, then,
f(x) = x^4-x^3-6x^2+4x+8
f(1) = (1)^4-(1)^3-6(1)^2+4(1)+8
f(1) = 6
We don't get a result of 0, which means x = 1 is not a root or zero of the function.

Trying x = -1 leads to:
f(x) = x^4-x^3-6x^2+4x+8
f(-1) = (-1)^4-(-1)^3-6(-1)^2+4(-1)+8
f(-1) = 0
We get zero this time, so x = -1 is a root of f(x)

Repeat these steps for the remaining possible roots listed above. You should find that only the following are roots: -2, -1, 2
Side note: x = 2 is a double root

Answer: -2, -1, and 2 are zeros of f(x)

RELATED QUESTIONS
Use the remainder theorem to find the remainder when f(x) is divided by g(x)... (answered by ramkikk66)

Use the remainder theorem to find the remainder when f(x) is divided by x+3. Then use the (answered by sachi)

Use the remainder theorem to find the remainder when 4x^23-3x^13+2x^3-3 is divided by... (answered by Theo)

Use the remainder theorem to find the remainder when f(x) is divided by x+3. Then use the (answered by josgarithmetic)

please help use the remainder theorem to determine the remainder when {{{3t^2+5t-7}}} is (answered by gsmani_iyer)

3. Use the remainder theorem to find the remainder when f(x) is divided by x-2 . Then use (answered by lwsshak3)

3. Use the remainder theorem to find the remainder when f(x) is divided by x-2 . Then (answered by lwsshak3)

use the reminder theorem to find the remainder when the given polynomial is divided by... (answered by The Troll Kid)