Question

actorize each of the following polynomials.

Answer 1

Solution:-

given by::- (i)

${x}^{3} - 2 {x}^{2} - 5x + 6 \\ {x}^{3} - {x}^{2} - {x}^{2} + x - 6x + 6 \\ {x}^{2} (x - 1) - x(x - 1) - 6(x - 1) \\ (x - 1)( {x}^{2} - x - 6) \\ (x - 1)( {x}^{2} - 3x + 2x - 6) \\ (x - 1)(x(x - 3) + 2(x - 6)) \\ (x - 1)(x - 3)(x - 2) \: ans$
(ii)
$4 {x}^{3} - 7x + 3 \\ 4 {x}^{3} - 4 {x}^{2} + 4 {x}^{2} - 7x + 3 \\ {4x}^{3} - {4x}^{2} + 4 {x}^{2} - 4x - 3x + 3 \\ 4 {x}^{2} (x - 1) + 4x(x - 1) - 3(x - 1) \\ (x - 1)(4 {x}^{2} + 4x - 3) \\ (x - 1)( {4x}^{2} + 6x - 2x - 3) \\ (x - 1)(2x(2x + 3) - 1(2x + 3)) \\ (x - 1)(2x + 3)(2x - 1) \: ans$

(iii)
${x}^{3} - 23 {x}^{2} + 142x - 120 \\ {x}^{3} - {x}^{2} - 22 {x}^{2} + 22x + 120x - 120 \\ {x}^{2} (x - 1) - 22x(x - 1) + 120(x - 1) \\ (x - 1)( {x}^{2} - 22x + 120) \\ (x - 1)( {x}^{2} - 12x - 10x + 120) \\ (x - 1)(x(x - 12) - 10(x - 12)) \\ (x - 1)(x - 12)(x - 10) \: ans$
(iv)
$4 {x}^{3} - 5 {x}^{2} + 7x - 6 \\ 4 {x}^{3} - 4 {x}^{2} - {x}^{2} + x + 6x - 6 \\ 4 {x}^{2} (x - 1) - x(x - 1) + 6(x - 1) \\ (x - 1)(4 {x}^{2} - x + 6)ans$
(v)
${x}^{3} - 7x + 6 \\ {x}^{3} - {x}^{2} + {x}^{2} - x - 6x + 6 \\ {x}^{2} (x - 1) + x(x - 1) - 6(x - 1) \\ (x - 1)( {x}^{2} + x - 6) \\ (x - 1)( {x}^{2} + 3x - 2x - 6) \\ (x - 1)(x(x + 3) - 2(x + 3)) \\ (x - 1)(x + 3)(x - 2) \: ans$
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Answer 2

DIVISION PROCESS IN THE ATTACHMENT .

SOLUTION :
i) Given : x³ - 2 x² - 5 x + 6
Let f(x) = x³ - 2 x² - 5 x + 6
The factors of the constant term 6 is ±1,±2,±3.

Put x = 1
f(1) = 1³ - 2 (1)² - 5 (1) + 6
= 1 - 2 - 5 + 6
= 7 - 7 = 0
Hence, (x - 1) is a factor of f(x).
To find other two factors we divide f(x) by (x-1).
x³ - 2 x² - 5 x + 6 / (x-1)
= (x² - x - 6)
x³ - 2 x² - 5 x + 6 = (x² - x - 6) (x-1)
x³ - 2 x² - 5 x + 6 = (x² - 3x +2x - 6) (x-1)
= [x(x -3) +2(x-3)] (x-1)
[by splitting the middle term]
x³ - 2 x² - 5 x + 6 = (x - 3) (x + 2) (x -1)

Hence, the required factors of the cubic Polynomial are (x - 1) (x - 3) (x + 2)

ii)
Let f (x) = 4 x³ - 7 x + 3
The factors of the constant term 3 is ±1,±3.
Put, x = 1in f(x)
f(1) = 4 (1)³ -7 (1) + 3
= 4 - 7 + 3 = 7 - 7 = 0
Hence, (x - 1) is a factor of f(x).
To find other two factors we divide f(x) by x-1.
4 x³ - 7 x + 3 / x-1 = 4 x² + 4 x - 3
4 x³ - 7 x + 3 = ( 4 x² + 4 x - 3) (x -1)
4 x³ - 7 x + 3 = ( 4 x² + 6 x - 2x - 3) (x -1)
[by splitting the middle term]
= 2x(2x + 3) -1(2x +3) (x -1)
4 x³ - 7 x + 3 = (2x +3) (2 x - 1) (x -1)

Hence, the required factors of the cubic Polynomial are (2x +3) (2 x - 1) (x -1)


iii) Let f (x) = x³ - 23 x² + 142 x - 120
The factors of the constant term 120 is ±1,±2,±3,±4,±5,±6,±,10,±12...
Put, x = 1in f(x)
f (1) = 1³ - 23 (1)² + 142 (1) - 120
= 1 - 23 + 142 - 120 = 143 - 143 = 0

Hence, (x - 1) is a factor of f(x).
To find other two factors we divide f(x) by x-1.

x³ - 23 x² + 142 x - 120 / (x - 1) = x² - 22 x + 120

x³ - 23 x² + 142 x - 120 =( x² - 22 x + 120)(x -1)
x³ - 23 x² + 142 x - 120 =[( x² - 12x -10 x + 120)] (x -1)
[ by middle term splitting]
x³ - 23 x² + 142 x - 120 = x(x - 12) -10(x -12) (x -1)
x³ - 23 x² + 142 x - 120 = (x - 12) (x -10) (x -1)

Hence, the required factors of the cubic Polynomial are (x - 1) (x - 12) (x - 10)

iv)
Let f (x) = 4 x³ - 5 x² + 7 x - 6
The factors of the constant term 6 is ±1,±2,±3.
Put x = 1,in f(x)
f (1) = 4 (1)³ - 5 (1)² + 7 (1) - 6
= 4 - 5 + 7 - 6 = 11 - 11 = 0
Hence, (x - 1) is a factor of f(x).
To find other two factors we divide f(x) by x-1.
4 x³ - 5 x² + 7 x - 6 / (x -1) = 4 x² - x + 6
This quadratic equation (4 x² - x + 6) can not be further factorise.
Hence, the required factors of the cubic Polynomial are (x -1) (4 x² - x + 6)

v)
Let f (x) = x³ - 7 x + 6
The factors of the constant term 6 is ±1,±2,±3.
Put x = 1,in f(x)
f (1) = (1)³ - 7 (1) + 6
= 1 - 7 + 6 = 7 - 7 = 0
Hence, (x - 1) is a factor of f(x).
To find other two factors we divide f(x) by x-1.
x³ - 7 x + 6 / (x -1) = x² + x - 6
x³ - 7 x + 6 = (x - 1) (x² + x - 6)
[factorising (x² + x - 6) by middle term splitting]
x³ - 7 x + 6 = (x - 1) (x² + 3x -2x - 6)
x³ - 7 x + 6 = (x - 1) (x(x +3) - 2(x + 3))
x³ - 7 x + 6 = (x - 1) (x-2) (x+3)

Hence, the required factors of the cubic Polynomial are (x - 1) (x-2) (x+3)

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