Question

use long division to find the quotient. show all work.
9 (10x² + 19x - 25)÷(x + 3)
10 (x³ - 19x - 30)÷(x - 5)

Explanation:

Step1: Divide leading terms

Divide the leading term of the dividend $10x^{2}+19x - 25$ (which is $10x^{2}$) by the leading term of the divisor $x + 3$ (which is $x$). $\frac{10x^{2}}{x}=10x$.

Step2: Multiply and subtract

Multiply $x + 3$ by $10x$ to get $10x^{2}+30x$. Subtract this from the dividend: $(10x^{2}+19x - 25)-(10x^{2}+30x)=10x^{2}+19x - 25 - 10x^{2}-30x=- 11x - 25$.

Step3: Repeat division

Divide the leading term of the new dividend $-11x - 25$ (which is $-11x$) by the leading term of the divisor $x$. $\frac{-11x}{x}=-11$.

Step4: Multiply and find remainder

Multiply $x + 3$ by $-11$ to get $-11x-33$. Subtract this from $-11x - 25$: $(-11x - 25)-(-11x-33)=-11x - 25 + 11x + 33 = 8$.
So, $(10x^{2}+19x - 25)\div(x + 3)=10x-11+\frac{8}{x + 3}$.

Now for $(x^{3}-19x - 30)\div(x - 5)$:

Step1: Divide leading terms

Divide the leading term of the dividend $x^{3}-19x - 30$ (which is $x^{3}$) by the leading term of the divisor $x - 5$ (which is $x$). $\frac{x^{3}}{x}=x^{2}$.

Step2: Multiply and subtract

Multiply $x - 5$ by $x^{2}$ to get $x^{3}-5x^{2}$. Subtract this from the dividend: $(x^{3}-19x - 30)-(x^{3}-5x^{2})=x^{3}-19x - 30 - x^{3}+5x^{2}=5x^{2}-19x - 30$.

Step3: Divide leading terms again

Divide the leading term of the new dividend $5x^{2}-19x - 30$ (which is $5x^{2}$) by the leading term of the divisor $x$. $\frac{5x^{2}}{x}=5x$.

Step4: Multiply and subtract

Multiply $x - 5$ by $5x$ to get $5x^{2}-25x$. Subtract this from $5x^{2}-19x - 30$: $(5x^{2}-19x - 30)-(5x^{2}-25x)=5x^{2}-19x - 30 - 5x^{2}+25x = 6x-30$.

Step5: Divide leading terms one - more time

Divide the leading term of the new dividend $6x-30$ (which is $6x$) by the leading term of the divisor $x$. $\frac{6x}{x}=6$.

Step6: Multiply and find remainder

Multiply $x - 5$ by $6$ to get $6x-30$. Subtract this from $6x-30$: $(6x-30)-(6x-30)=0$.
So, $(x^{3}-19x - 30)\div(x - 5)=x^{2}+5x + 6$.

Answer:

9] $10x-11+\frac{8}{x + 3}$
10] $x^{2}+5x + 6$