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

Formula

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

Introduction

$x$ is a variable, $a$ and $b$ are two constants. The sum of literals $x$ and $a$ formed a binomial $x+a$, and difference of the literals $x$ and $b$ formed another binomial $x-b$. The first term of both multinomials is same. So, they are special binomials. Therefore, the product of them can be called as the special binomials product and it is written in mathematical form as follows.

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

The product of the special polynomials $x+a$ and $x-b$ is equal to an algebraic expression $x^2+(a-b)x-ab$. This algebraic expression is used as expansion of the special binomial product $(x+a)(x-b)$ in algebraic mathematics.

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

Usage

This rule is actually used as a formula in mathematics if two opposite sign binomials contains one same variable, involved in multiplication.

Example

$(l+4)(l-7)$ $\,=\,$ $l^2+(4-7)l-(4 \times 7)$

$\implies$ $(l+4)(l-7)$ $\,=\,$ $l^2-3l-28$

In fact, the product of special binomials can also obtained by the multiplication of algebraic expressions but it takes more time than using this formula. Hence, the expansion of $(x+a)(x-b)$ identity is used as formula to get their product quickly in mathematics.

Proofs

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

Algebraic Method

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

Geometric Method

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

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