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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