Questions: Use synthetic division to divide. (5x^2-79) ÷ (x+4) (5x^2-79) ÷ (x+4) =

Transcript text: Use synthetic division to divide. \[ \left(5 x^{2}-79\right) \div(x+4) \] \[ \left(5 x^{2}-79\right) \div(x+4)= \]

Solution

Solution Steps

Step 1: Perform Synthetic Division

To divide the polynomial \(5x^2 - 79\) by the binomial \(x + 4\), we start by dividing the leading term of the dividend \(5x^2\) by the leading term of the divisor \(x\), which gives us \(5x\).

Step 2: Multiply and Subtract

Next, we multiply \(5x\) by the entire divisor \(x + 4\): \[ 5x \cdot (x + 4) = 5x^2 + 20x \] We then subtract this result from the original polynomial: \[ (5x^2 - 79) - (5x^2 + 20x) = -20x - 79 \]

Step 3: Repeat the Process

Now, we divide the new leading term \(-20x\) by the leading term of the divisor \(x\), resulting in \(-20\). We multiply \(-20\) by the divisor: \[ -20 \cdot (x + 4) = -20x - 80 \] Subtracting this from \(-20x - 79\) gives: \[ (-20x - 79) - (-20x - 80) = 1 \]

Step 4: Compile the Results

The quotient from our division is \(5x - 20\) and the remainder is \(1\). Therefore, we can express the result of the division as: \[ \frac{5x^2 - 79}{x + 4} = 5x - 20 + \frac{1}{x + 4} \]