Question

1. $(5x^3 - 60x^2 + 102x - 15) div (x - 10)$

Explanation:

Step1: Divide leading terms

$\frac{5x^3}{x}=5x^2$

Step2: Multiply divisor by $5x^2$

$5x^2(x-10)=5x^3-50x^2$

Step3: Subtract from dividend

$(5x^3-60x^2)-(5x^3-50x^2)=-10x^2$
Bring down $102x$: $-10x^2+102x$

Step4: Divide new leading terms

$\frac{-10x^2}{x}=-10x$

Step5: Multiply divisor by $-10x$

$-10x(x-10)=-10x^2+100x$

Step6: Subtract from current polynomial

$(-10x^2+102x)-(-10x^2+100x)=2x$
Bring down $-15$: $2x-15$

Step7: Divide leading terms

$\frac{2x}{x}=2$

Step8: Multiply divisor by 2

$2(x-10)=2x-20$

Step9: Subtract to find remainder

$(2x-15)-(2x-20)=5$

Answer:

Quotient: $5x^2 - 10x + 2$, Remainder: $5$
Or written as: $5x^2 - 10x + 2 + \frac{5}{x-10}$