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The expression $x^2-4x+5$ is a factor of $ax^3+bx^2+25$. Express the sum a+b as an integer.

Please give an explanation of how the answer

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$$\frac{a x^3+b x^2+25}{x^2-4x+5}=4a+b+a x+\frac{(11a+4b)x-5(4a+b)+25}{x^2-4x+5}$$ – non-expert Jan 4 '11 at 3:12
Please don't use the imperative mode; you aren't assigning homework to the group. If this is homework (seems likely, given the phrasing), then please use the [homework] tag. This is not linear algebra, so I've changed the tag. Finally, you should say what you have tried and why or how you are stuck. – Arturo Magidin Jan 4 '11 at 3:13
Arturo, the sentence " Please don't use the imperative mode" is a nice example of logical antinomy :-) (But I agree with you, of course) – Georges Elencwajg Jan 4 '11 at 8:41

2 Answers 2

up vote 3 down vote accepted

$\rm\ 0 = a\ x^3 + b\ x^2 + 25 - (x^2 - 4\ x + 5)\ (a\ x + 5) = (4\ a + b - 5)\ x^2 + (20 - 5\ a)\ x\ \Rightarrow \ a,\: b = \ldots$

Note: $\rm\ a\ x + 5\ $ comes from comparing leading and constant coefficients.

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If $x^2-4x+5$ is a factor of $ax^3+bx^2+25$, then $ax^3+bx^2+25=(x^2-4x+5)(\text{something})$. Since $ax^3+\cdots$ is a polynomial of degree 3 and $x^2-\cdots$ is a polynomial of degree 2, the "something" must be a polynomial of degree 1: $$ax^3+bx^2+0x+25=(x^2-4x+5)(\underline{\;\;\;\;\;\;}x+\underline{\;\;\;\;\;\;})$$ Try to fill in the two blanks based the terms on the left side that don't have $a$ and $b$ in them (for example, how will $+25$ end up in the product?), then finish the multiplication to find the values of $a$ and $b$.

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