We will discuss here about the expansion of (x + a)(x + b)(x + c).
(x + a)(x + b)(x + c) = (x + a){(x + b)(x + c)}
= (x + a){x\(^{2}\) + (b + c)x + bc}
= x{x\(^{2}\) + (b + c)x + bc} + a{x\(^{2}\) + (b + c)x + bc}
= x\(^{3}\) + (b + c)x\(^{2}\) + bcx + ax\(^{2}\) + a(b + c)x + abc
= x\(^{3}\) + (a + b + c)x\(^{2}\) + (bc + ab + ac)x + abc
= x\(^{3}\) + (a + b + c)x\(^{2}\) + (ab + bc + ca)x + abc
Therefore, (x + a)(x + b)(x + c) = x\(^{3}\) + (Sum of the constant terms)x\(^{2}\) + (Sum of the product of constant terms taking two at a time)x + Product of constant terms.
Solved Examples on Expansion of (x + a)(x + b)(x + c)
1. Find product of (x + 1)(x + 2)(x + 3)
Solution:
We know that, (x + a)(x + b)(x + c) = x\(^{3}\) + (a + b + c)x\(^{2}\) + (ab + bc + ca)x + abc
Here, a = 1, b = 2 and c = 3
Therefore, the product = x\(^{3}\) + (1 + 2 + 3)x\(^{2}\) + (1 ∙ 2 + 2 ∙ 3 + 3 ∙ 1)x + 1 ∙ 2 ∙ 3
= x\(^{3}\) + 6x\(^{2}\) + 11x + 6.
2. Find product of (x + 4)(x - 5)(x - 6)
Solution:
We know that, (x + a)(x + b)(x + c) = x\(^{3}\) + (a + b + c)x\(^{2}\) + (ab + bc + ca)x + abc
Here, a = 4, b = -5 and c = -6
Therefore, the product = x\(^{3}\) + {4 + (- 5) + (- 6)}x\(^{2}\) + {4 ∙ (-5) + (-5) ∙ (-6) + (-6) ∙ 4}x + 4 ∙ (-5) ∙ (-6)
= x\(^{3}\) + (4 - 5 – 6)x\(^{2}\) + (-20 + 30 – 24)x + 120.
= x\(^{3}\) - 7x\(^{2}\) - 14x + 120.
Problem on Expansion of (x + a)(x + b)(x + c)
1. Simplify the following by using standard formula and obtain the coefficients of x\(^{2}\) and x.
(i) (x + 1)(x + 3)(x + 5)
(ii) (a + 2)(a – 4)(a + 6)
(iii) (2x + 1)(2x + 3)(2x + 5)
Answers:
1. (i) x\(^{3}\) + 9x\(^{2}\) + 23x + 15
(ii) a\(^{3}\) + 4a\(^{2}\) – 20a - 48
(iii) 8x\(^{3}\) + 36x\(^{3}\) + 46x + 15
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