Question

What is the formula of \[\left( x-a \right)\left( x-b \right)=?\] in exponents?

Answer 1

Hint: To write the given expression in terms of exponents, expand the given algebraic expression by multiplying the terms in each of the brackets. Simplify the terms to rewrite the given expression.

Complete step-by-step solution -
We have the expression \[\left( x-a \right)\left( x-b \right)\]. We have to write this expression in terms of exponents.
We will expand the expression by multiplying the terms in each of the brackets.
By expanding the given expression, we can rewrite \[\left( x-a \right)\left( x-b \right)\] as \[\left( x-a \right)\left( x-b \right)=x\left( x-b \right)-a\left( x-b \right)\].
Further simplifying the above equation, we have \[\left( x-a \right)\left( x-b \right)=x\left( x-b \right)-a\left( x-b \right)={{x}^{2}}-bx-ax+ab\].
We can rewrite the above equation as \[\left( x-a \right)\left( x-b \right)={{x}^{2}}-bx-ax+ab={{x}^{2}}-\left( a+b \right)x+ab\].
Hence, the expression \[\left( x-a \right)\left( x-b \right)\] can be written in terms of exponents as \[\left( x-a \right)\left( x-b \right)={{x}^{2}}-\left( a+b \right)x+ab\].
We observe that \[\left( x-a \right)\left( x-b \right)\] is an algebraic expression. However, we observe that \[\left( x-a \right)\left( x-b \right)={{x}^{2}}-\left( a+b \right)x+ab\] is an algebraic identity. An algebraic identity is an equality that holds for all possible values of its variables. We can prove each of the identities by performing basic algebraic operations such as addition, multiplication, subtraction and division. However, an algebraic expression is an expression built from integer constants, variables and algebraic operations. An algebraic expression differs from an algebraic identity in the way that the value of an algebraic expression changes with the change in variables. However, an algebraic identity is an equality which holds for all possible values of variables.

Note: We can verify that the algebraic identity \[\left( x-a \right)\left( x-b \right)={{x}^{2}}-\left( a+b \right)x+ab\] holds from for all values of ‘x’ by using the substitution method. In this method, we substitute the values for the variables and perform arithmetic operations. We can also prove an algebraic identity using an activity method, in which we use geometry to prove the algebraic identity.