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# Homework Help?

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Hi everyone!

I have three problems that I didnt quite understand on my homework:

1. When x^4 + 6x^3 - ax^2 - 45x -15 is divided by x^2 - x - 6 the remainder is 3. What is a?

2. Find the remainder when r^14 is divided by r - 1.

3. Find a polynomial q(x) such that (x+1)^3  + x^2 times q(x) has degree less than 2.

Thanks so much if you can help!

Dec 12, 2019

#1
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1.
When $$x^4 + 6x^3 - ax^2 - 45x -15$$ is divided by $$x^2 - x - 6$$ the remainder is $$3$$.
What is $$a$$?

$$\text{Let x^2 - x - 6 = (x-3)(x+2)}$$

$$\begin{array}{|lrcll|} \hline & \mathbf{\dfrac{x^4 + 6x^3 - ax^2 - 45x -15}{(x-3)(x+2)}} &=& \mathbf{q(x) + \dfrac{3}{(x-3)(x+2)}} \quad | \quad \times (x-3)(x+2) \\\\ x=3: & x^4 + 6x^3 - ax^2 - 45x -15 &=& q(x)(x-3)(x+2) + 3 \\ & 3^4+6(3^3)-a(3^2)-45(3)-15 &=& q(x)(0)(5) + 3 \\ & 3^4+6(3^3)-a(3^2)-45(3)-15 &=& 3 \\ & 81+162-9a-135-15 &=& 3 \\ & 93-9a &=& 3 \quad | \quad :3 \\ & 31-3a &=& 1 \\ & 3a &=& 31-1 \\ & 3a &=& 30 \quad | \quad :3 \\ & \mathbf{a} &=& \mathbf{10} \\ \hline \end{array}$$

Dec 12, 2019
edited by heureka  Dec 12, 2019
edited by heureka  Dec 12, 2019
#2
+23905
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2.

Find the remainder when r^14 is divided by r - 1.

The remainder is 1

Dec 12, 2019