Question

1. ((x^{3}+4x^{2}+4x)div(x + 2))\\8. ((2x^{4}+5x^{3}+9x^{2}+10x + 8)div(x^{2}+4))\\9. ((6x^{3}-13x^{2}-12x + 4)div(2x + 1))\\10. ((2x^{6}+5x^{5}-3x^{3})-(4x^{6}-3x^{5}-7x^{4}))\\11. ((x^{4}-6x^{2}-5)div(x + 2))\\12. ((3x^{4}+6x^{3}-5x^{2})+(2x^{4}-7x^{2}-9x^{3}))

Explanation:

Problem 7:

Step1: Factor numerator first

$x^3 + 4x^2 + 4x = x(x^2 + 4x + 4) = x(x+2)^2$

Step2: Cancel common factor

$\frac{x(x+2)^2}{x+2} = x(x+2)$

Step3: Expand simplified form

$x(x+2) = x^2 + 2x$

Problem 8:

Step1: Divide term by term via polynomial long division

Divide $2x^4$ by $x^2$: $2x^2$
Multiply $(x^2+4)$ by $2x^2$: $2x^4 + 8x^2$
Subtract from dividend: $(2x^4+5x^3+9x^2+10x+8)-(2x^4+8x^2)=5x^3+x^2+10x+8$

Step2: Next term of quotient

Divide $5x^3$ by $x^2$: $5x$
Multiply $(x^2+4)$ by $5x$: $5x^3 + 20x$
Subtract: $(5x^3+x^2+10x+8)-(5x^3+20x)=x^2-10x+8$

Step3: Final term of quotient

Divide $x^2$ by $x^2$: $1$
Multiply $(x^2+4)$ by $1$: $x^2+4$
Subtract: $(x^2-10x+8)-(x^2+4)=-10x+4$

Step4: Combine quotient and remainder

Quotient: $2x^2+5x+1$, Remainder: $-10x+4$

Problem 9:

Step1: Polynomial long division first term

Divide $6x^3$ by $2x$: $3x^2$
Multiply $(2x+1)$ by $3x^2$: $6x^3+3x^2$
Subtract: $(6x^3-13x^2-12x+4)-(6x^3+3x^2)=-16x^2-12x+4$

Step2: Next quotient term

Divide $-16x^2$ by $2x$: $-8x$
Multiply $(2x+1)$ by $-8x$: $-16x^2-8x$
Subtract: $(-16x^2-12x+4)-(-16x^2-8x)=-4x+4$

Step3: Final quotient term

Divide $-4x$ by $2x$: $-2$
Multiply $(2x+1)$ by $-2$: $-4x-2$
Subtract: $(-4x+4)-(-4x-2)=6$

Step4: Combine quotient and remainder

Quotient: $3x^2-8x-2$, Remainder: $6$

Problem 10:

Step1: Distribute the negative sign

$2x^6+5x^5-3x^4-4x^6+3x^5+7x^4$

Step2: Combine like terms

$(2x^6-4x^6)+(5x^5+3x^5)+(-3x^4+7x^4) = -2x^6+8x^5+4x^4$

Problem 11:

Step1: Polynomial long division first term

Divide $x^4$ by $x$: $x^3$
Multiply $(x+2)$ by $x^3$: $x^4+2x^3$
Subtract: $(x^4-6x^2-5)-(x^4+2x^3)=-2x^3-6x^2-5$

Step2: Next quotient term

Divide $-2x^3$ by $x$: $-2x^2$
Multiply $(x+2)$ by $-2x^2$: $-2x^3-4x^2$
Subtract: $(-2x^3-6x^2-5)-(-2x^3-4x^2)=-2x^2-5$

Step3: Next quotient term

Divide $-2x^2$ by $x$: $-2x$
Multiply $(x+2)$ by $-2x$: $-2x^2-4x$
Subtract: $(-2x^2-5)-(-2x^2-4x)=4x-5$

Step4: Final quotient term

Divide $4x$ by $x$: $4$
Multiply $(x+2)$ by $4$: $4x+8$
Subtract: $(4x-5)-(4x+8)=-13$

Step5: Combine quotient and remainder

Quotient: $x^3-2x^2-2x+4$, Remainder: $-13$

Problem 12:

Step1: Remove parentheses

$3x^4+6x^3-5x^2+2x^4-7x^2-9x^3$

Step2: Combine like terms

$(3x^4+2x^4)+(6x^3-9x^3)+(-5x^2-7x^2)=5x^4-3x^3-12x^2$

Answer:

1. $\boldsymbol{x^2 + 2x}$
2. $\boldsymbol{2x^2+5x+1+\frac{-10x+4}{x^2+4}}$
3. $\boldsymbol{3x^2-8x-2+\frac{6}{2x+1}}$
4. $\boldsymbol{-2x^6+8x^5+4x^4}$
5. $\boldsymbol{x^3-2x^2-2x+4+\frac{-13}{x+2}}$
6. $\boldsymbol{5x^4-3x^3-12x^2}$