Factorize the following polynomials-i x2-x2-8x2-x-12ii x-52-5 x-25-24iii x2-6 x2-8x2-6 x-8-64iv x2-2 x-3x2-2 x-5-35v y-2y-3y-8y-3-56vi y2-5 yy2-5 y-2-24vii x-3x-42x-5-6

(i) (x2 nbsp; ndash; nbsp;x)2 nbsp; ndash; 8(x2 nbsp; ndash; nbsp;x) + 12 Let nbsp;x2 nbsp; ndash; nbsp;x = nbsp;z. #8756;x2 x2 8x ...

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Byju's Answer

Standard IX

Mathematics

Algebraic Identities

Factorize the...

Question

Factorize the following polynomials.
(i) (x² – x)² – 8 (x² – x) + 12
(ii) (x – 5)² – (5x – 25) – 24
(iii) (x² – 6x)² – 8 (x² – 6x + 8) – 64
(iv) (x² – 2x + 3) (x² – 2x + 5) – 35
(v) (y + 2) (y – 3) (y + 8) (y + 3) + 56
(vi) (y² + 5y) (y² + 5y – 2) – 24
(vii) (x – 3) (x – 4)² (x – 5) – 6

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Solution

(i)
(x² – x)² – 8(x² – x) + 12
Let x² – x = z.
$∴ {(x^{2} - x)}^{2} - 8 (x^{2} - x) + 12 = z^{2} - 8 z + 12 = z^{2} - 6 z - 2 z + 12 = z (z - 6) - 2 (z - 6) = (z - 6) (z - 2)$
$= (x^{2} - x - 6) (x^{2} - x - 2) (Replace z = x^{2} - x) = (x^{2} - 3 x + 2 x - 6) (x^{2} - 2 x + x - 2) = [x (x - 3) + 2 (x - 3)] [x (x - 2) + 1 (x - 2)] = (x - 3) (x + 2) (x - 2) (x + 1)$
(ii)
(x – 5)² – (5x – 25) – 24
= (x – 5)² – 5(x – 5) – 24
Let x – 5 = z.
$∴ {(x - 5)}^{2} - 5 (x - 5) - 24 = z^{2} - 5 z - 24 = z^{2} - 8 z + 3 z - 24 = z (z - 8) + 3 (z - 8) = (z - 8) (z + 3)$
$= (x - 5 - 8) (x - 5 + 3) (Replace z = x - 5) = (x - 13) (x - 2)$
(iii)
(x² – 6x)² – 8(x² – 6x + 8) – 64
= (x² – 6x)² – 8(x² – 6x) – 64 – 64
= (x² – 6x)² – 8(x² – 6x) – 128
Let x² – 6x = z.
$∴ {(x^{2} - 6 x)}^{2} - 8 (x^{2} - 6 x) - 128 = z^{2} - 8 z - 128 = z^{2} - 16 z + 8 z - 128 = z (z - 16) + 8 (z - 16) = (z - 16) (z + 8)$
$= (x^{2} - 6 x - 16) (x^{2} - 6 x + 8) (Replace z = x^{2} - 6 x) = (x^{2} - 8 x + 2 x - 16) (x^{2} - 4 x - 2 x + 8) = [x (x - 8) + 2 (x - 8)] [x (x - 4) - 2 (x - 4)] = (x - 8) (x + 2) (x - 4) (x - 2)$

(iv)
(x² – 2x + 3)(x² – 2x + 5) – 35
Let x² – 2x = z.
$∴ (x^{2} - 2 x + 3) (x^{2} - 2 x + 5) - 35 = (z + 3) (z + 5) - 35 = z^{2} + 5 z + 3 z + 15 - 35 = z^{2} + 8 z - 20 = z^{2} + 10 z - 2 z - 20$
$= z (z + 10) - 2 (z + 10) = (z + 10) (z - 2) = (x^{2} - 2 x + 10) (x^{2} - 2 x - 2) (Replace z = x^{2} - 2 x)$
(v)
(y + 2)(y – 3)(y + 8)(y + 3) + 56
= (y + 2)(y + 3)(y + 8)(y – 3) + 56
= (y² + 5y + 6)(y² + 5y – 24) + 56
Let y² + 5y = z.
$∴ (y^{2} + 5 y + 6) (y^{2} + 5 y - 24) + 56 = (z + 6) (z - 24) + 56 = z^{2} - 18 z - 144 + 56 = z^{2} - 18 z - 88$
$= z^{2} - 22 z + 4 z - 88 = z (z - 22) + 4 (z - 22) = (z - 22) (z + 4) = (y^{2} + 5 y - 22) (y^{2} + 5 y + 4) (Replace z = y^{2} + 5 y)$
$= (y^{2} + 5 y - 22) (y^{2} + 4 y + y + 4) = (y^{2} + 5 y - 22) [y (y + 4) + 1 (y + 4)] = (y^{2} + 5 y - 22) (y + 4) (y + 1)$
(vi)
(y² + 5y)(y² + 5y – 2) – 24
Let y² + 5y = z.
$∴ (y^{2} + 5 y) (y^{2} + 5 y - 2) - 24 = z (z - 2) - 24 = z^{2} - 2 z - 24 = z^{2} - 6 z + 4 z - 24 = z (z - 6) + 4 (z - 6) = (z - 6) (z + 4)$
$= (y^{2} + 5 y - 6) (y^{2} + 5 y + 4) (Replace z = y^{2} + 5 y) = (y^{2} + 6 y - y - 6) (y^{2} + 4 y + y + 4) = [y (y + 6) - 1 (y + 6)] [y (y + 4) + 1 (y + 4)] = (y + 6) (y - 1) (y + 4) (y + 1)$
(vii)
(x – 3)(x – 4)²(x – 5) – 6
= (x – 3)(x – 5)(x – 4)² – 6
= (x² – 8x + 15)(x² – 8x + 16) – 6
Let x² – 8x = z.
$∴ (x^{2} - 8 x + 15) (x^{2} - 8 x + 16) - 6 = (z + 15) (z + 16) - 6 = z^{2} + 31 z + 240 - 6 = z^{2} + 31 z + 234$
$= z^{2} + 18 z + 13 z + 234 = z (z + 18) + 13 (z + 18) = (z + 18) (z + 13) = (x^{2} - 8 x + 18) (x^{2} - 8 x + 13) (Replace z = x^{2} - 8 x)$

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