Expansion of (x + a)(x + b)(x + c)

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





9th Grade Math

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