+20 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!
+26367 1.
When \(x^4 + 6x^3 - ax^2 - 45x -15\) is divided by \(x^2 - x - 6\) the remainder is \(3\).
What is \(a\)?
\(\text{Let $x^2 - x - 6 = (x-3)(x+2)$}\)
\(\begin{array}{|lrcll|} \hline & \mathbf{\dfrac{x^4 + 6x^3 - ax^2 - 45x -15}{(x-3)(x+2)}} &=& \mathbf{q(x) + \dfrac{3}{(x-3)(x+2)}} \quad | \quad \times (x-3)(x+2) \\\\ x=3: & x^4 + 6x^3 - ax^2 - 45x -15 &=& q(x)(x-3)(x+2) + 3 \\ & 3^4+6(3^3)-a(3^2)-45(3)-15 &=& q(x)(0)(5) + 3 \\ & 3^4+6(3^3)-a(3^2)-45(3)-15 &=& 3 \\ & 81+162-9a-135-15 &=& 3 \\ & 93-9a &=& 3 \quad | \quad :3 \\ & 31-3a &=& 1 \\ & 3a &=& 31-1 \\ & 3a &=& 30 \quad | \quad :3 \\ & \mathbf{a} &=& \mathbf{10} \\ \hline \end{array}\)
