factorize x^3+5x^2-2x-24
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Answer:
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).
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