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Factorisation Using Algebraic Identities

If px= x 2 ...

Question

If $p (x) = x^{2} - 2 x + 1$ , $q (x) = x^{4} - 1$ and $r (x) = x^{3} - 2 x^{2} - 5 x + 6$ , find their HCF.

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Solution

We factorise all of them:
$p (x) = x^{2} - 2 x + 1 = {(x - 1)}^{2}$ ,
$q (x) = x^{4} - 1 = (x^{2} - 1) (x^{2} + 1) = (x - 1) (x + 1) (x^{2} + 1)$ ,
$r (x) = x^{3} - 2 x^{2} - 5 x + 6 = x^{3} - x^{2} + x - 6 x + 6$
$= x^{2} (x - 1) - x (x - 1) - 6 (x - 1)$
$= (x - 1) (x^{2} - x - 6) = (x - 1) (x - 3) (x + 2)$ .
Observe that $H C F (p (x), q (x)) = x - 1$ and $H C F (x - 1, r (x)) = x - 1$ . Hence the HCF of $p (x), q (x), r (x)$ is $x - 1$ .

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Similar questions

If dividend = $x^{4} + x^{3} - 2 x^{2} + x + 1$ and divisor = $(x - 1)$ , find the quotient, $q (x)$ and remainder $r (x)$ .

p (x)

g (x)

q (x)

r (x)

p (x)

p (x) = (x^{3} - 27) (x^{2} - 3 x + 2)

q (x) = (x^{2} + 3 x + 9) (x^{2} - 5 x + 6)

Now for each pair of polynomials p(x) and q(x) given below, find p(x) + q(x) and p(x) − q(x). Also compute p(1) + q(1) and p(1) − q(1) for each.

(i)

(ii)

(iii)

(iv)

(v)

p (x) = x^{4} - 2 x^{3} - 15 x^{2}

q (x) = x^{3} - 9 x

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