Constant in Polynomial Equation
Find the constant $ a $ such that \[(x^2 - 3x + 4)(2x^2 +ax + 7) = 2x^4 -11x^3 +30x^2 -41x +28.\]
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- +
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- $\frac{a}{b}$
- .
- 0
- =
- %
- $a^n$
- $a^{\circ}$
- $a_n$
- $\sqrt{}$
- $\sqrt[n]{}$
- $\pi$
- $\ln{}$
- $\log$
- $\theta$
- $\sin{}$
- $\cos{}$
- $\tan{}$
- $($
- $)$
- $[$
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- $\cap$
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- $,$
- $\infty$
Solution
We look at the coefficient of $ x $ in the expansion of the product on the left. We get an $ x $ term when we multiply $ (+4)(+ax) $ and when we multiply $ (-3x)(+7) $ in the expansion. So, on the left the $ x $ term is $ 4ax -21x $. Since this term must equal $ -41x $, we have $ 4ax -21x = -41x $, so $ a = \boxed{-5} $.
We can check our answer (and check that it is indeed possible to find a solution to this problem) by multiplying out the left when $ a=-5 $: \begin{align*}
(x^2&-3x+4)(2x^2-5x+7)\\
&= x^2(2x^2-5x+7) -3x(2x^2-5x+7) + 4(2x^2-5x+7)\\ &=2x^4 -11x^3 +30x^2 -41x +28.\end{align*}
This matches the polynomial given in the problem, so our answer is correct.
Freeflows