Question

irections: divide each polynomials. use y 1. $(x^3 - 5x^2 - 10x + 2) div (x + 2)$

Explanation:

Step1: Use Polynomial Long Division

Divide the first term of the dividend \(x^3 - 5x^2 - 10x + 2\) by the first term of the divisor \(x + 2\). So, \(\frac{x^3}{x}=x^2\).
Multiply the divisor \(x + 2\) by \(x^2\): \(x^2(x + 2)=x^3 + 2x^2\).
Subtract this from the dividend: \((x^3 - 5x^2 - 10x + 2)-(x^3 + 2x^2)=-7x^2 - 10x + 2\).

Step2: Divide the New First Term

Now, divide the first term of \(-7x^2 - 10x + 2\) by \(x\): \(\frac{-7x^2}{x}=-7x\).
Multiply the divisor \(x + 2\) by \(-7x\): \(-7x(x + 2)=-7x^2 - 14x\).
Subtract this from \(-7x^2 - 10x + 2\): \((-7x^2 - 10x + 2)-(-7x^2 - 14x)=4x + 2\).

Step3: Divide the New First Term Again

Divide the first term of \(4x + 2\) by \(x\): \(\frac{4x}{x}=4\).
Multiply the divisor \(x + 2\) by \(4\): \(4(x + 2)=4x + 8\).
Subtract this from \(4x + 2\): \((4x + 2)-(4x + 8)=-6\).

Answer:

The result of the division is \(x^2 - 7x + 4-\frac{6}{x + 2}\)