Question

How do you express the product of $2{x^2} + 7x - 10$ and $x + 5$ in standard form?

Answer 1

Hint: In this question, we are given two expressions and we have been asked to express the product in the standard form. First, multiply the two expressions and bring them to the simplest form by adding or subtracting the like terms. Once the terms have been simplified, arrange them in such a way that their powers are in decreasing order.

Complete step-by-step solution:
The answer to this question lies in the product of the two expressions. Let us multiply the two given expressions.
$ \Rightarrow \left( {x + 5} \right)\left( {2{x^2} + 7x - 10} \right)$
We will distribute the first expression such that each term of the first expression can be multiplied with every term of second expression.
$ \Rightarrow x\left( {2{x^2} + 7x - 10} \right) + 5\left( {2{x^2} + 7x - 10} \right)$
Now, we will multiply the bracket term, and we get
$ \Rightarrow 2{x^3} + 7{x^2} - 10x + 10{x^2} + 35x - 50$
Next step is to group the like terms. (Like terms are those terms which have same variables and their powers, for example, $45{x^4}$ and $\dfrac{{23}}{7}{x^4}$ are like terms)
$ \Rightarrow 2{x^3} + \left( {35x - 10x} \right) + \left( {10{x^2} + 7{x^2}} \right) - 50$
Now, we will add or subtract the like terms.
$ \Rightarrow 2{x^3} + 25x + 17{x^2} - 50$
This is the product of the given two expressions. We have been asked to find the product in standard form. In our case, we need to find the standard form of writing a cubic equation in one variable. It is defined as follows:
$ \Rightarrow a{x^3} + b{x^2} + cx + d = 0$
Let us write our equation in the standard form.
 $ \Rightarrow 2{x^3} + 25x + 17{x^2} - 50$
On rewriting we get
$ \Rightarrow 2{x^3} + 17{x^2} + 25x - 50$

Hence, our answer is $2{x^3} + 17{x^2} + 25x - 50$.

Note: What is standard form? Standard form is a way of writing down a particular equation or set of numbers. We can simply define the standard form of a cubic equation in one variable as – the power of the variable goes on decreasing, that is, the highest power comes first, followed by a power smaller than that, ending at the lowest power.