Question

Using factor theorem,factorize the polynomial:

1) x⁴-2x³-7x^​2+8x+12

Answer 1

Consider the following polynomial.
f(x)=x4+2x3−7x2−8x+12
The constant term is 12
Therefore the possible zeroes are ±1, ±2, ±3, ±4 ±6, ±12.
Put x=1 in f(x)
f(1)=(1)4+2(1)3−7(1)2−8(1)+12
=(1)+2(1)−7(1)−8(1)+12
=1+2−7−8+12
=15−15
=0
Since f(1)=0,
therefore by factor theorem (x−1) is a factor of f(x)

Put x=−3 in f(x)
f(−3)=(−3)4+2(−3)3−7(−3)2−8(−3)+12 =81+2(−27)−7(9)+24+12
=81−54−63+24+12
=117−117
=0
Since f(−3)=0,
therefore by factor theorem (x+3) is a factor of f(x)

Put x=2 in f(x)
f(2)=(2)4+2(2)3−7(2)2−8(2)+12
=16+2(8)−7(4)−16+12
=16+16−28−16+12
=44−44
=0
Since f(2)=0,
therefore by factor theorem (x−2) is a factor of f(x)

Put x=−2 in f(x)
f(−2)=(−2)4+2(−2)3−7(−2)2−8(−2)+12 =16+2(−8)−7(4)+16+12
=16−16−28+16+12
=−28+28
=0
Since f(−2)=0,
therefore by factor theorem (x+2) is a factor of f(x)

Hence the required factors of given polynomial are
f(x)=(x−1)(x+3)(x+2)(x−2)