Questions: Use synthetic division to perform the division. x^4-3x^3-13x^2+16x-5 / x-5

Solution Steps

Divide \( x^{4} \) by \( x \), resulting in \( x^{3} \). The remaining polynomial is \( 2 x^{3} - 13 x^{2} + 16 x - 5 \).

Divide \( 2 x^{3} \) by \( x \), resulting in \( 2 x^{2} \). The remaining polynomial is \( -3 x^{2} + 16 x - 5 \).

Divide \( -3 x^{2} \) by \( x \), resulting in \( -3 x \). The remaining polynomial is \( x - 5 \).

Divide \( x \) by \( x \), resulting in \( 1 \). The remaining polynomial is \( 0 \).

The quotient is \( x^{3} + 2 x^{2} - 3 x + 1 \) and the remainder is \( 0 \).

The complete division expression is given by: \[ \frac{x^{4} - 3 x^{3} - 13 x^{2} + 16 x - 5}{x - 5} = x^{3} + 2 x^{2} - 3 x + 1 \]

Final Answer