Use synthetic division to divide. (2x^3-6x+8x^2-24)/(x+4) square xneq -4

To use synthetic division to divide \( \frac{2x^3 - 6x + 8x^2 - 24}{x+4} \), we first need to rewrite the polynomial in standard form and then perform the synthetic division.<br /><br />The polynomial in standard form is:<br />\[ 2x^3 + 8x^2 - 6x - 24 \]<br /><br />We will divide this by \( x + 4 \). In synthetic division, we use the root of the divisor \( x + 4 \), which is \( -4 \).<br /><br />Set up the synthetic division:<br /><br />1. Write down the coefficients of the polynomial: \( 2, 8, -6, -24 \).<br />2. Use \( -4 \) as the divisor.<br /><br />\[<br />\begin{array}{r|rrrr}<br />-4 & 2 & 8 & -6 & -24 \\<br /> & & -8 & 0 & 24 \\<br />\hline<br /> & 2 & 0 & -6 & 0 \\<br />\end{array}<br />\]<br /><br />Here’s the step-by-step process:<br /><br />1. Bring down the first coefficient (2).<br />2. Multiply \( -4 \) by the value just written below the line (2), giving \( -8 \).<br />3. Add this result to the next coefficient (8), giving \( 0 \).<br />4. Multiply \( -4 \) by the new value (0), giving \( 0 \).<br />5. Add this result to the next coefficient (-6), giving \( -6 \).<br />6. Multiply \( -4 \) by the new value (-6), giving \( 24 \).<br />7. Add this result to the next coefficient (-24), giving \( 0 \).<br /><br />The final row gives the coefficients of the quotient polynomial and the remainder. Since the remainder is 0, the quotient polynomial is:<br />\[ 2x^2 + 0x - 6 = 2x^2 - 6 \]<br /><br />Thus, the result of the division is:<br />\[ \frac{2x^3 + 8x^2 - 6x - 24}{x + 4} = 2x^2 - 6 \quad \text{for} \quad x \neq -4 \]