Questions: Use synthetic division to divide f(x)=x^3+18x^2+101x+180 by x+4. Use the result to find all zeros of f. (x^3+18x^2+101x+180) ÷ (x+4) = (Do not factor. If there is a remainder, type your answer in the form quotient + remainder / divisor )

Use synthetic division to divide f(x)=x^3+18x^2+101x+180 by x+4. Use the result to find all zeros of f.

(x^3+18x^2+101x+180) ÷ (x+4) =

(Do not factor. If there is a remainder, type your answer in the form quotient + remainder / divisor )

Transcript text: Use synthetic division to divide $f(x)=x^{3}+18 x^{2}+101 x+180$ by $x+4$. Use the result to find all zeros of $f$. \[ \left(x^{3}+18 x^{2}+101 x+180\right) \div(x+4)=\square \] (Do not factor. If there is a remainder, type your answer in the form quotient $+\frac{\text { remainder }}{\text { divisor }}$ )

Solution

Solution Steps

Solution Approach

Use synthetic division to divide the polynomial $ f(x) = x^3 + 18x^2 + 101x + 180 $ by $ x + 4 $.
The result of the synthetic division will give us the quotient and the remainder.
Use the quotient to find the zeros of $ f(x) $.

Step 1: Perform Synthetic Division

We will divide the polynomial $ f(x) = x^3 + 18x^2 + 101x + 180 $ by $ x + 4 $ using synthetic division. The coefficients of $ f(x) $ are $ [1, 18, 101, 180] $ and the divisor is $ -4 $.

After performing synthetic division, we find the quotient to be $ 1x^2 + 14x + 45 $ and the remainder is $ 0 $.

Step 2: Write the Result

The result of the division can be expressed as: \[ f(x) = (x + 4)(1x^2 + 14x + 45) \]

Step 3: Find the Zeros of the Quotient

To find the zeros of $ f(x) $, we need to solve the equation: \[ 1x^2 + 14x + 45 = 0 \] Using the quadratic formula $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $, where $ a = 1 $, $ b = 14 $, and $ c = 45 $: \[ x = \frac{-14 \pm \sqrt{14^2 - 4 \cdot 1 \cdot 45}}{2 \cdot 1} \] Calculating the discriminant: \[ 14^2 - 4 \cdot 1 \cdot 45 = 196 - 180 = 16 \] Thus, we have: \[ x = \frac{-14 \pm \sqrt{16}}{2} = \frac{-14 \pm 4}{2} \] This gives us two solutions: \[ x_1 = \frac{-10}{2} = -5 \quad \text{and} \quad x_2 = \frac{-18}{2} = -9 \]