Question

Factorize the expression $40 + 3x - {x^2}$ .

Answer 1

Hint: Given polynomial is of degree 2. Polynomials of degree 2 are known as Quadratic polynomials. Quadratic polynomials can be factored by the help of splitting the middle term method. In this method, the middle term is split into two terms in such a way that the polynomial remains unchanged and becomes easy to factorise.

Complete step-by-step solution:
For factorising the given quadratic polynomial $40 + 3x - {x^2}$ , we can use the splitting method in which the middle term is split into two terms such that the sum of the terms gives us the original middle term and product of the terms gives us the product of the constant term and coefficient of ${x^2}$.
So, $40 + 3x - {x^2}$
First, we will take the negative sign common from the bracket so that the coefficient of ${x^2}$ becomes positive. So, we get,
$ = $$ - \left( { - 40 - 3x + {x^2}} \right)$
Rearranging the terms in the descending order of the power of the variable x so as to simplify the expression.
$ = $$ - \left( {{x^2} - 3x - 40} \right)$
Now, we split the middle term $ - 3x$ into two terms $ - 8x$ and $5x$ since the product of these two terms, $ - 40{x^2}$ is equal to the product of the constant term and coefficient of ${x^2}$ and sum of these terms gives us the original middle term, $ - 3x$.
$ = $$ - \left( {{x^2} - 8x + 5x - 40} \right)$
Taking out x common from first two brackets and $ - 6$ common from last two brackets, we get,
$ = $$ - \left( {x\left( {x - 8} \right) + 5\left( {x - 8} \right)} \right)$
Taking out $\left( {x - 8} \right)$ common from the terms, we get,
$ = $$ - \left( {x + 5} \right)\left( {x - 8} \right)$
So, the factored form of the quadratic polynomial $40 + 3x - {x^2}$ is $ - \left( {x + 5} \right)\left( {x - 8} \right)$.
Additional Information: Similarly quadratic polynomials, quadratic equations can also be solved using factorisation methods. Besides factorisation, there are various methods to solve quadratic equations such as completing the square method and using the Quadratic formula.

Note: Splitting of middle term can be a tedious process at times when the product of the constant term and coefficient of ${x^2}$ is a large number with a large number of divisors. Special care should be taken in such cases.