Question

homework assignment 5.4 dividing polynomials
due friday by 11:59pm points 8 submitting an external tool
homework assignment 5.4 dividing polynomials
score: 3/8 answered: 3/8
question 4
rewrite
\\(\frac{25x^{4}+25x^{3}+41x^{2}+20x + 10}{5x^{2}+2x + 2}\\) in the form
\\(q(x)+\frac{r(x)}{5x^{2}+2x + 2}\\).
\\(\frac{25x^{4}+25x^{3}+41x^{2}+20x + 10}{5x^{2}+2x + 2}=\square+\frac{\square}{5x^{2}+2x + 2}\\)
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Explanation:

Step1: Divide leading - terms

Divide $25x^{4}$ by $5x^{2}$, we get $5x^{2}$.

Step2: Multiply and subtract

Multiply $5x^{2}+2x + 2$ by $5x^{2}$: $5x^{2}(5x^{2}+2x + 2)=25x^{4}+10x^{3}+10x^{2}$.
Subtract from the dividend: $(25x^{4}+25x^{3}+41x^{2}+20x + 10)-(25x^{4}+10x^{3}+10x^{2})=15x^{3}+31x^{2}+20x + 10$.

Step3: Divide new leading - term

Divide $15x^{3}$ by $5x^{2}$, we get $3x$.

Step4: Multiply and subtract again

Multiply $5x^{2}+2x + 2$ by $3x$: $3x(5x^{2}+2x + 2)=15x^{3}+6x^{2}+6x$.
Subtract from the new dividend: $(15x^{3}+31x^{2}+20x + 10)-(15x^{3}+6x^{2}+6x)=25x^{2}+14x + 10$.

Step5: Divide new leading - term

Divide $25x^{2}$ by $5x^{2}$, we get $5$.

Step6: Multiply and find the remainder

Multiply $5x^{2}+2x + 2$ by $5$: $5(5x^{2}+2x + 2)=25x^{2}+10x + 10$.
Subtract from the new dividend: $(25x^{2}+14x + 10)-(25x^{2}+10x + 10)=4x$.

Answer:

$5x^{2}+3x + 5+\frac{4x}{5x^{2}+2x + 2}$