Factorization of Cubic Polynomial
In this section, we will discuss Factorization of Cubic Polynomial.The degree of the cubic (highest exponent) polynomial is 3.
There are different identities in Factorization of Cubic Polynomial .
They are as follows :
1) a^{3} + b^{3} + 3a^{2}b + 3b^{2}a = (a + b)^{3}
2) a^{3} - b^{3} - 3a^{2}b + 3b^{2}a = (a - b)^{3}
3) a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})
4) a^{3}- b^{3} = (a - b)(a^{2} + ab + b^{2})
5) a^{3} + b^{3}+ c^{3}- 3abc
= (a + b + c)(a^{2} + b^{2}+ c^{2} - ab - bc - ac)
6) If a + b + c = 0 then a^{3} + b^{3}+ c^{3}=3abc
Using Identities
Examples :
1)27x ^{3} - 8
Solution :
As first term and the 2nd term are perfect cube, we use the identity of cube
(a^{3} - b^{3}) = ( a - b)(a^{2} +ab + b^{2}) |
a = 3x and b = 2
27x ^{3} - 8 = (3x -2)[(3x) ^{2} + 6x + 2 ^{2} )]
= (3x -2)(9x ^{2} + 6x + 4)are the factors.
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2) 8x ^{3} +1
Solution :
As first term and the 2nd term are perfect cube, we use the identity of cube
(a^{3} + b^{3}) = ( a + b)(a^{2} -ab + b^{2}) |
a = 2x and b = 1
8x ^{3} + 1 = (2x + 1)[(2x) ^{2} - 2x + 1 ^{2} )]
= (2x +1 )(4x ^{2} - 2x + 1)are the factors.
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3) 27x ^{3} + y ^{3} + 9x ^{2} y + 3xy ^{2}
Solution :
As in the given equation there are two perfect cubes, so we can use 1st identity.
27x ^{3} + y ^{3} + 27x ^{2} y + 9xy ^{2}
= (3x) ^{3} + (y) ^{3} + 3(3x) ^{2} y + 3(3x)y ^{2}
= (3x + y) ^{3} [ here a = 3x and b= y;a ^{3} + b ^{3} + 3a ^{2} b + 3b ^{2} a = (a + b) ^{3} ]
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4) (1.5) ^{3} - (0.9) ^{3} - (0.6) ^{3}
Solution :
(1.5) ^{3} - (0.9) ^{3} + (-0.6) ^{3}
Let a = 1.5, b = - 0.9 and c = -0.6
As a + b + c = 1.5 + 0.9 - 0.6 = 0
a ^{3} + b ^{3} + c ^{3} = 3abc
= 3(1.5)(- 0.9)(-0.6)
= 2.430
Factoring
• Factorization by common factor
• Factorization by Grouping
• Factorization using Identities
• Factorization of Cubic Polynomial
• Solved Examples on Factorization