Question

question 6 (3 points)
what is the remainder for the following quotient?
$(x^3 - 5x^2 - 12x + 36) \div (x^2 - 3x - 18)$
\\(\bigcirc\\) a \quad 72
\\(\bigcirc\\) b \quad 0
\\(\bigcirc\\) c \quad x - 6
\\(\bigcirc\\) d \quad x - 2

question 7 (2 points)
given polynomial function f and a zero of f, find the other zeros.
$f(x) = 4x^3 - 25x^2 - 154x + 40$ ; 10
zeros = \underline{qquad}, \underline{qquad}
blank 1:
blank 2:

question 8 (2 points)
given polynomial function f and a zero of f, find the other zeros.
$f(x) = x^3 - 2x^2 - 21x - 18$ ; $-3$
zeros = \underline{qquad}, \underline{qquad}
blank 1:
blank 2:

Explanation:

Question 6

Step1: Perform polynomial long division

Divide \(x^3 - 5x^2 - 12x + 36\) by \(x^2 - 3x - 18\).
First term: \(\frac{x^3}{x^2}=x\). Multiply divisor by \(x\): \(x(x^2 - 3x - 18)=x^3 - 3x^2 - 18x\).
Subtract from dividend: \((x^3 - 5x^2 - 12x + 36)-(x^3 - 3x^2 - 18x)= -2x^2 + 6x + 36\).

Step2: Next term in quotient

\(\frac{-2x^2}{x^2}=-2\). Multiply divisor by \(-2\): \(-2(x^2 - 3x - 18)= -2x^2 + 6x + 36\).

Step3: Subtract and find remainder

Subtract: \((-2x^2 + 6x + 36)-(-2x^2 + 6x + 36)=0\).
So the remainder is \(0\).

Step1: Use synthetic division with root 10

Divide \(4x^3 - 25x^2 - 154x + 40\) by \(x - 10\).
Coefficients: \(4, -25, -154, 40\); root \(10\).
Bring down \(4\). Multiply by \(10\): \(40\). Add to \(-25\): \(15\).
Multiply \(15\) by \(10\): \(150\). Add to \(-154\): \(-4\).
Multiply \(-4\) by \(10\): \(-40\). Add to \(40\): \(0\).
Quotient polynomial: \(4x^2 + 15x - 4\).

Step2: Factor the quadratic

Factor \(4x^2 + 15x - 4\): \(4x^2 + 16x - x - 4 = 4x(x + 4) - 1(x + 4)=(4x - 1)(x + 4)\).

Step3: Find roots

Set factors to zero: \(4x - 1 = 0 \Rightarrow x=\frac{1}{4}\); \(x + 4 = 0 \Rightarrow x=-4\).

Step1: Use synthetic division with root -3

Divide \(x^3 - 2x^2 - 21x - 18\) by \(x + 3\) (root \(-3\)).
Coefficients: \(1, -2, -21, -18\); root \(-3\).
Bring down \(1\). Multiply by \(-3\): \(-3\). Add to \(-2\): \(-5\).
Multiply \(-5\) by \(-3\): \(15\). Add to \(-21\): \(-6\).
Multiply \(-6\) by \(-3\): \(18\). Add to \(-18\): \(0\).
Quotient polynomial: \(x^2 - 5x - 6\).

Step2: Factor the quadratic

Factor \(x^2 - 5x - 6\): \(x^2 - 6x + x - 6 = x(x - 6) + 1(x - 6)=(x - 6)(x + 1)\).

Step3: Find roots

Set factors to zero: \(x - 6 = 0 \Rightarrow x=6\); \(x + 1 = 0 \Rightarrow x=-1\).

Answer:

b. 0

Question 7