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GCD of Polynomials

For what valu...

Question

For what values of $a$ and $b$ the polynomials $p (x) = (x^{2} + 3 x + 2) (x^{2} + 2 x + a)$ and $q (x) = (x^{2} + 7 x + 12) (x^{2} + 7 x + b)$ have $(x + 3) (x + 4)$ as their HCF?

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Solution

We observe
$x^{2} + 3 x + 2 = (x + 1) (x + 2)$ and $x^{2} + 7 x + 12 = (x + 3) (x + 4)$ .
Thus
$p (x) = (x + 1) (x + 2) (x^{2} + 2 x + a)$ , $q (x) = (x + 3) (x + 4) (x^{2} + 7 x + b)$
Hence $x + 3$ divides $x^{2} + 2 x + a$ and $x + 1$ divides $x^{2} + 7 x + b$ . Write
$x^{2} + 2 x + a = (x + 3) f (x)$ , $x^{2} + 7 x + b = (x + 1) (g (x)$ , for some polynomials $f (x)$ and $g (x)$ . Taking $x = - 3$ in the first relation above, we get ${(- 3)}^{2} + 2 \times (- 3) + a = 0$ . We get $a = - 3$ . Taking $x = - 1$ in the second relation, we get ${(- 1)}^{2} + 7 \times (- 1) + b = 0$ . We get $b = 6$ .

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