Question

How do you simplify $7rs\left( 4{{r}^{2}}+9{{s}^{3}}-7rs \right)?$

Answer 1

Hint: We will use the distributive property to simplify the problem. We will multiply the term outside the brackets with the terms inside the brackets. Then we will get the simplified form of the given polynomial expression.

Complete step-by-step answer:
Let us consider the given polynomial expression, $7rs\left( 4{{r}^{2}}+9{{s}^{3}}-7rs \right).$
We are asked to simplify the given polynomial expression.
For that, we will use the distributive property of multiplication over addition and subtraction. We will multiply the term $7rs$ with each of the terms inside the brackets.
We know that the distributive property is given by $a\left( b+c \right)=ab+ac.$
Also, we are able to write the same property for subtraction.
And that equation is given by $a\left( b-c \right)=ab-ac.$
Let us apply these properties in our problem.
We will get $7rs\left( 4{{r}^{2}}+9{{s}^{3}}-7rs \right)=7rs\cdot 4{{r}^{2}}+7rs\cdot 9{{s}^{3}}-7rs\cdot 7rs.$
Let us do the required multiplications in the obtained form of the polynomial.
We know that the coefficients will be multiplied in the usual way. The multiplication of the variables will be done as follows: $rs\cdot {{r}^{2}}={{r}^{3}}s, rs\cdot {{s}^{3}}=r{{s}^{4}}$ and $rs\cdot rs={{\left( rs \right)}^{2}}={{r}^{2}}{{s}^{2}}.$
So, we can apply this in the obtained polynomial expression.
As a result, we will get the polynomial expression as the following one after the multiplication \[7rs\left( 4{{r}^{2}}+9{{s}^{3}}-7rs \right)=28{{r}^{3}}s+63r{{s}^{4}}-49{{r}^{2}}{{s}^{2}}.\]
Hence the simplified form of the given polynomial expression is \[7rs\left( 4{{r}^{2}}+9{{s}^{3}}-7rs \right)=28{{r}^{3}}s+63r{{s}^{4}}-49{{r}^{2}}{{s}^{2}}.\]

Note: We should always remember that the multiplication is distributive over the operations addition and subtraction. Although subtraction is the same as addition, that is subtraction is the addition of negative numbers, we usually mention it separately.