Factorize the following expressions - i 5x-1x2-4 5x-1x-4-x 0ii 36x2-12x-1-25y2iii x4-x2-1

(i) ( 5x 1x)^2+4( 5x 1x)+4 =( 5x 1x)^2+2×( 5x 1x)× 2+2^2 =( 5x 1x+2 )^2 (ii) 36x^2 12x+1 25y^2 =(6x)^2 2× 6x× 1+1^2 (5y)^2 =(6x 1)^2 (5y)^2 ={(6x 1) 5y}{( ...

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Byju's Answer

Standard IX

Mathematics

Factorisation Using Algebraic Identities

Factorize the...

Question

Factorize the following expressions :
(i) ${(5 x - \frac{1}{x})}^{2} + 4 (5 x - \frac{1}{x}) + 4, x \neq 0$
(ii) $36 x^{2} - 12 x + 1 - 25 y^{2}$

(iii) $x^{4} + x^{2} + 1$

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Solution

(i) ${(5 x - \frac{1}{x})}^{2} + 4 (5 x - \frac{1}{x}) + 4$

$= {(5 x - \frac{1}{x})}^{2} + 2 \times (5 x - \frac{1}{x}) \times 2 + 2^{2}$

$= {(5 x - \frac{1}{x} + 2)}^{2}$

(ii) $36 x^{2} - 12 x + 1 - 25 y^{2}$

$= {(6 x)}^{2} - 2 \times 6 x \times 1 + 1^{2} - {(5 y)}^{2}$

$= {(6 x - 1)}^{2} - {(5 y)}^{2}$

$= {(6 x - 1) - 5 y} {(6 x - 1) + 5 y}$

$= (6 x - 1 - 5 y) (6 x - 1 + 5 y)$

$= (6 x - 5 y - 1) (6 x + 5 y - 1)$

(iii) $x^{4} + x^{2} + 1 = (x^{4} + 2 x^{2} + 1) - x^{2}$

$= {(x^{2} + 1)}^{2} - x^{2} = (x^{2} + 1 - x) (x^{2} + 1 + x)$
$= (x^{2} - x + 1) (x^{2} + x + 1)$

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Similar questions

Now in each of the pairs of polynomials given below, check whether the first is a factor of the second:

(i) x + 1, x³ − 1

(ii) x − 1, x³ + 1

(iii) x + 1, x³ + 1

(iv) x² − 1, x⁴ − 1

(v) x − 1, x⁴ − 1

(vi) x + 1, x⁴ − 1

(vii) x − 2, x² − 5x + 1

(viii) x + 2, x² + 5x + 6

(ix)

(x) 1.3x − 2.6, x² − 5x + 6

Determine which of the following polynomials have $(x + 1)$ as a factor: [2 MARKS]
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(ii) $x^{4} + x^{3} + x^{2} + x + 1$

If $(x + \frac{1}{x}) = 4$ , find the value of
(i) If $(x^{2} + \frac{1}{x^{2}})$ and
(ii) $(x^{4} + \frac{1}{x^{4}})$

Factorize the following expression:

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Q. Integrate the following expression with respect to

x

\frac{1}{x^{4} + x^{2} + 1}

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Factorize the following expressions : (i) (5x−1x)2+4(5x−1x)+4,x≠0 (ii) 36x2−12x+1−25y2 (iii) x4+x2+1

Factorize the following expressions :
(i) ${(5 x - \frac{1}{x})}^{2} + 4 (5 x - \frac{1}{x}) + 4, x \neq 0$
(ii) $36 x^{2} - 12 x + 1 - 25 y^{2}$

(iii) $x^{4} + x^{2} + 1$