(x+a)(x+b)(x+c)(x+d) Formula

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Jul 03, 2024

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Formula (x+a)(x+b)(x+c)(x+d)

Introduction

In algebra, the formula (x+a)(x+b)(x+c)(x+d) is a product of four binomials. This formula is used to expand the product of four expressions, each consisting of a variable x and a constant term. In this article, we will explore the formula, its expansion, and some examples to illustrate its application.

The Formula

The formula is:

(x+a)(x+b)(x+c)(x+d)

Where x is a variable and a, b, c, and d are constants.

Expansion of the Formula

To expand the formula, we need to follow the order of operations (PEMDAS) and multiply each binomial with the other:

(x+a)(x+b)(x+c)(x+d) = (x^2 + (a+b)x + ab)(x^2 + (c+d)x + cd)

Then, we multiply the two resulting binomials:

= x^4 + (a+b+c+d)x^3 + (ab + bc + cd + ad + ac + bd)x^2 + (abc + abd + acd + bcd)x + abcd

This is the expanded form of the formula.

Examples

Let's consider some examples to illustrate the application of the formula:

Example 1

Find the expansion of (x+2)(x+3)(x+4)(x+5)

Using the formula, we get:

(x+2)(x+3)(x+4)(x+5) = x^4 + (2+3+4+5)x^3 + (23 + 34 + 45 + 24 + 25 + 35)x^2 + (234 + 235 + 245 + 345)x + 234*5 = x^4 + 14x^3 + 59x^2 + 130x + 120

Example 2

Find the expansion of (x-1)(x+2)(x-3)(x+4)

Using the formula, we get:

**(x-1)(x+2)(x-3)(x+4) = x^4 + (-1+2-3+4)x^3 + (-12 + 2(-3) + (-3)4 + (-1)4 + (-1)2 + 2(-3))x^2 + (-12(-3) + (-1)24 + (-1)*(-3)4 + 2(-3)4)x + (-1)2(-3)4 = x^4 + 2x^3 - 23x^2 - 16x + 24

These examples demonstrate the application of the formula (x+a)(x+b)(x+c)(x+d) to expand the product of four binomials.

Conclusion

In conclusion, the formula (x+a)(x+b)(x+c)(x+d) is a powerful tool for expanding the product of four binomials. By applying this formula, we can simplify complex algebraic expressions and solve problems in various fields, such as mathematics, physics, and engineering.