$(x-a)(x-b)$ identity

Formula

$(x-a)(x-b)$ $\,=\,$ $x^2-(a+b)x+ab$

Introduction

$a$ and $b$ are two constants, and $x$ is a variable. The literals $x$ and $a$ formed a binomial $x-a$ by their difference. Similarly, the literals $x$ and $b$ also formed another binomial $x-b$ by their difference. The two polynomials are called as special binomials because the first term in each expression is same.

Hence, the product of them is called as the special product of binomials. It is written in the following mathematical form algebraically.

$(x-a)(x-b)$

The product of them is equal to an algebraic expression $x^2-(a+b)x+ab$. Therefore, the product of special binomials $x-a$ and $x-b$ is expanded as $x^2-(a+b)x+ab$ in algebraic mathematics.

$(x-a)(x-b)$ $\,=\,$ $x^2-(a+b)x+ab$

Use

This rule is used as a formula in mathematics when two difference basis binomials are involved in multiplication and a term in both polynomials is same.

Example

$(p-3)(p-5)$ $\,=\,$ $p^2-(3+5)p+(3 \times 5)$

$\implies$ $(p-3)(p-5)$ $\,=\,$ $p^2-8p+15$

Actually, the product $(p-3)(p-5)$ can be obtained as per multiplication of algebraic expressions but it is a time taking process. Therefore, the algebraic identity is used as a formula in mathematics for obtaining the product of two difference basis special binomials in only two steps.

Proofs

The expansion of $(x-a)(x-b)$ can be derived in mathematical form in two different mathematical approaches.

Algebraic Method

Learn how to derive the special product $(x-a)(x-b)$ by multiplying the binomials $x-a$ and $x-b$.

Geometric Method

Learn how to derive the special product of binomials $(x-a)$ and $(x-b)$ by the area of a square.

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