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Factor, Remainder and Factorisation

Factorise: ...

Question

Factorise: $(x^{2} - 5 x) (x^{2} - 5 x - 20) + 84$

A

(x - 6) (x + 1) (x - 7) (x + 2)

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B

(x - 6) (x - 1) (x - 7) (x + 2)

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C

(x - 6) (x + 1) (x - 7) (x - 2)

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D

(x - 6) (x - 3) (x - 7) (x + 2)

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Solution

The correct option is A $(x - 6) (x + 1) (x - 7) (x + 2)$
$(x^{2} - 5 x) (x^{2} - 5 x - 20) + 84 = (x^{2} - 5 x) (x^{2} - 5 x) - 20 (x^{2} - 5 x) + 84$
$= {(x^{2} - 5 x)}^{2} - 20 (x^{2} - 5 x) + 84$
Let $x^{2} - 5 x = a$ . Then, the given expression
$= a^{2} - 20 a + 84 = a^{2} - 16 a - 14 a + 84$
$= a (a - 6) - 14 (a - 6) = (a - 6) (a - 14)$
$= (x^{2} - 5 x - 6) (x^{2} - 5 x - 14) = (x^{2} - 6 x - x + 6) (x^{2} - 7 x + 2 x - 16)$
$= [x (x - 6) + 1 (x - 6)] [x (x - 7) + 2 (x - 7)]$
$= (x - 6) (x + 1) (x - 7) (x + 2)$

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Similar questions

Q. The factors of x³ − 7x + 6 are

(a) x (x − 6) (x − 1)

(b) (x² − 6) (x − 1)

(c) (x + 1) (x + 2) (x + 3)

(d) (x − 1) (x + 3) (x − 2)

Q. Factorise:
(i)

12 x^{2} - 7 x - 1

2 x^{2} + 7 x + 3

6 x^{2} + 5 x - 6

3 x^{2} - x - 4

Q. Which of the following are quadratic equations?

(i) x² + 6x − 4 = 0
(ii)

\sqrt{3} x^{2} - 2 x + \frac{1}{2} = 0

x^{2} + \frac{1}{x^{2}} = 5

x - \frac{3}{x} = x^{2}

2 x^{2} - \sqrt{3 x} + 9 = 0

x^{2} - 2 x - \sqrt{x} - 5 = 0

(vii) 3x² − 5x + 9 = x² − 7x + 3
(viii)

x + \frac{1}{x} = 1

(ix) x² − 3x = 0
(x)

{(x + \frac{1}{x})}^{2} = 3 (x + \frac{1}{x}) + 4

(xi) (2x + 1) (3x + 2) = 6(x − 1) (x − 2)
(xii)

x + \frac{1}{x} = x^{2}, x \neq 0

(xiii) 16x² − 3 = (2x + 5) (5x − 3)
(xiv) (x + 2)³ = x³ − 4
(xv) x(x + 1) + 8 = (x + 2) (x − 2)

Q. Which of the following are quadratic equation in x ?
(i)

x^{2} - x + 3 = 0

2 x^{2} + \frac{5}{2} x - \sqrt{3} = 0

\sqrt{2} x^{2} + 7 x + 5 \sqrt{2} = 0

\frac{1}{3} x^{2} + \frac{1}{5} x - 2 = 0

x^{2} - 3 x - \sqrt{x} + 4 = 0

x - \frac{6}{x} = 3

x + \frac{2}{x} = x^{2}

x^{2} - \frac{1}{x^{2}} = 5

{(x + 2)}^{3} = x^{3} - 8

(2 x + 3) (3 x + 2) = 6 (x - 1) (x - 2)

{(x + \frac{1}{x})}^{2} = 2 (x + \frac{1}{x}) + 3

Q. Question 23
Factorise
(i)

x^{2} + 9 x + 18

6 x^{2} + 7 x - 3

2 x^{2} - 7 x - 15

84 - 2 r - 2 r^{2}

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