Question

factorise x cube minus 5 x squar +2x+8
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Answer 1

Answer:

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Step-by-step explanation:

Let the given polynomial be p(x)=x

3

−5x

2

−2x+24.

We will now substitute various values of x until we get p(x)=0 as follows:

Forx=0

p(0)=(0)

3

−5(0)

2

−(2×0)+24=0−0−0+24=24



=0

∴p(0)



=0

Forx=1

p(1)=(1)

3

−5(1)

2

−(2×1)+24=1−5−2+24=25−7=18



=0

∴p(1)



=0

Forx=−2

p(−2)=(−2)

3

−5(−2)

2

−(2×−2)+24=−8−20+4+24=28−28=0

∴p(−2)=0

Thus, (x+2) is a factor of p(x).

Now,

p(x)=(x+2)⋅g(x).....(1)

⇒g(x)=

(x+2)

p(x)

Therefore, g(x) is obtained by after dividing p(x) by (x+2) as shown in the above image:

From the division, we get the quotient g(x)=x

2

−7x+12 and now we factorize it as follows:

x

2

−7x+12

=x

2

−4x−3x+12

=x(x−4)−3(x−4)

=(x−3)(x−4)

From equation 1, we get p(x)=(x+2)(x−3)(x−4).

Hence, x

3

−5x

2

−2x+24=(x+2)(x−3)(x−4).

Answer 2

Question -

Factorise -

${x}^{3} - 5 {x}^{2} + 2x + 8$

Solution -

Let p(x) = x³-5x²+2x+8

Factors of 8 are

$\pm1, \: \pm2, \: \pm4, \:, \pm8$

By trial and error method -

p(x) = x³-5x²+2x+8

Putting p(x) = -1 we will get -

p(-1)= (-1)³ - 5(-1)²+2(-1)+8

p(-1)= -1-5-2+8

p(-1)= - 8+8

p(-1)= 0

We find that p(-1) =0, so (x+1) is a factor of p(x)

Now,

$= {x }^{3} - 5 {x}^{2} + 2x + 8$

$= {x}^{3} + {x}^{2} - 6 {x}^{2} - 6x + 8x + 8$

$= {x}^{2} (x + 1) - 6x(x + 1) + 8(x + 1)$

Taking (x+1) common

$= (x + 1)( {x}^{2} - 6x + 8)$

Now x² - 6x+85 can be factorised either by splitting the middle term factorisation or by using the factor theorem.

By splitting the middle term, we have

$= {x}^{2} - 6x + 8$

$= {x}^{2} - (4 + 2)x + 8$

$= {x}^{2} - 4x - 2x + 8$

$= x(x - 4) - 2(x - 4)$

Taking (x-4) common

$= (x - 4)(x - 2)$

Therefore x³-5x²+2x+8 = (x+1)(x-4)(x-2)