Questions: Use synthetic division to divide (f(x)=x^3-5 x^2+2 x+8) by (x+1). Use the result to find all zeros of (f). [ left(x^3-5 x^2+2 x+8right) div(x+1)= ] (Do not factor. If there is a remainder, type your answer in the form quotient (+fractext remainder text divisor ).)

$Use synthetic division to divide (f(x)=x^3-5 x^2+2 x+8) by (x+1). Use the result to find all zeros of (f). [ left(x^3-5 x^2+2 x+8right) div(x+1)= ] (Do not factor. If there is a remainder, type your answer in the form quotient (+fractext remainder text divisor ).)$

Transcript text: Use synthetic division to divide $f(x)=x^{3}-5 x^{2}+2 x+8$ by $x+1$. Use the result to find all zeros of $f$. \[ \left(x^{3}-5 x^{2}+2 x+8\right) \div(x+1)= \] $\square$ (Do not factor. If there is a remainder, type your answer in the form quotient $+\frac{\text { remainder }}{\text { divisor }}$.)

Solution

Solution Steps

Step 1: Perform Synthetic Division

We divide the polynomial $ f(x) = x^{3} - 5x^{2} + 2x + 8 $ by $ x + 1 $. The division process yields:

Divide $ x^{3} $ by $ x $, resulting in $ x^{2} $.
Multiply $ x^{2} $ by $ x + 1 $ to get $ x^{3} + x^{2} $.
Subtract $ (x^{3} + x^{2}) $ from $ f(x) $ to get the new polynomial: \[ -6x^{2} + 2x + 8 \]
Divide $ -6x^{2} $ by $ x $, resulting in $ -6x $.
Multiply $ -6x $ by $ x + 1 $ to get $ -6x^{2} - 6x $.
Subtract $ (-6x^{2} - 6x) $ from the current polynomial to get: \[ 8x + 8 \]
Divide $ 8x $ by $ x $, resulting in $ 8 $.
Multiply $ 8 $ by $ x + 1 $ to get $ 8x + 8 $.
Subtract $ (8x + 8) $ from the current polynomial to get a remainder of $ 0 $.

Thus, the quotient is $ x^{2} - 6x + 8 $ and the remainder is $ 0 $.

Step 2: Write the Division Result

The result of the division can be expressed as: \[ \frac{x^{3} - 5x^{2} + 2x + 8}{x + 1} = x^{2} - 6x + 8 \]

Step 3: Find the Zeros of the Quotient

To find the zeros of $ f(x) $, we first solve the equation $ x^{2} - 6x + 8 = 0 $. Factoring gives: \[ (x - 2)(x - 4) = 0 \] Thus, the zeros from the quotient are $ x = 2 $ and $ x = 4 $.

Step 4: Find the Zero of the Divisor

Next, we solve the equation $ x + 1 = 0 $ to find the zero of the divisor: \[ x = -1 \]

Step 5: Combine All Zeros

The complete set of zeros for the polynomial $ f(x) $ is: \[ \{2, 4, -1\} \]