Question

Expand the expression $\left( {a + 2} \right)\left( {a - 1} \right)$

Answer 1

Hint: The required expansion is obtained by using the algebraic identity $\left( {x + c} \right)\left( {x + b} \right) = {x^2} + \left( {c + b} \right)x + cb$. Comparing the identity with the given polynomial we get the value of $x,b$ and $c$. Substituting the obtained values in the identity we get the required expansion.

Complete step-by-step solution:
We are given two linear polynomials and we need to find the product of the polynomials
We can expand it using an algebraic identity
$\left( {x + c} \right)\left( {x + b} \right) = {x^2} + \left( {c + b} \right)x + cb$
Comparing this with the given polynomial
We get $x$ to be $a$ , $c$ to be $2$ and $b$ to be $\left( { - 1} \right)$
Substituting these values in the above identity
We get ,
$\left( {a + 2} \right)\left( {a - 1} \right) = {a^2} + \left( {2 + \left( { - 1} \right)} \right)a + \left( 2 \right)\left( { - 1} \right)$
Further simplifying we get ,
$
  \left( {a + 2} \right)\left( {a - 1} \right) = {a^2} + \left( {2 - 1} \right)a - 2 \\
  \left( {a + 2} \right)\left( {a - 1} \right) = {a^2} + a - 2 \\
$
Hence the required expansion is obtained .

Additional Information:
An expression with degree one is called a monomial or linear polynomial .When two linear polynomials are added or subtracted it results in a linear polynomial . But it is not necessary that the product of the linear polynomials must be linear, it may be quadratic , cubic or a polynomial of higher degree.

Note:
Alternative method :
We are given two linear polynomials and it can be expanded even with using the identity
The given expression can be expanded using the distributive property.
That is ,
$\left( {a + 2} \right)\left( {a - 1} \right)$ can be written as $a\left( {a - 1} \right) + 2\left( {a - 1} \right)$
$\left( {a + 2} \right)\left( {a - 1} \right) = a\left( {a - 1} \right) + 2\left( {a - 1} \right)$
Now let us multiply each term of this linear polynomial $\left( {a - 1} \right)$ by $a$ in the first part
$\left( {a + 2} \right)\left( {a - 1} \right) = {a^2} - a + 2\left( {a - 1} \right)$
Same way multiply each term of the polynomial $\left( {a - 1} \right)$ by $2$ in the second part
$\left( {a + 2} \right)\left( {a - 1} \right) = {a^2} - a + 2a - 2$
Further simplifying by adding the like terms we get
$\left( {a + 2} \right)\left( {a - 1} \right) = {a^2} + a - 2$
Hence the required expansion can be obtained either by using the algebraic identity or the distributive property