Question

How do you multiply the polynomial $({x^2} + 2x - 1)({x^2} + 2x + 5)$?

Answer 1

Hint: In this question, we need to multiply each other the given polynomial and simplify it. There are two pairs which are separated by parenthesis. Firstly, consider the terms in the first parenthesis. We take the first term in it and multiply to all the terms in the next parenthesis. Do the same for the remaining terms also. Then we add all the expressions obtained and combine the like terms and simplify it to obtain the solution.

Complete step by step solution:
Given an expression of the form $({x^2} + 2x - 1)({x^2} + 2x + 5)$ …… (1)
We are asked to multiply the above expression given in the equation (1).
Observe that there are two different polynomials which are separated by a parenthesis.
Note that the highest power in both the parenthesis is 2.
So after multiplying both the polynomials, we will obtain an expression of degree 4. i.e. highest power of the resultant polynomial will be 4.
Now consider the first polynomial given by, $({x^2} + 2x - 1)$
Now we multiply one by one term of this polynomial to the other polynomial which is $({x^2} + 2x + 5)$.
Now we multiply the first term of $({x^2} + 2x - 1)$, which is ${x^2}$ to each term of $({x^2} + 2x + 5)$, we get,
$ \Rightarrow {x^2} \cdot ({x^2} + 2x + 5)$
$ \Rightarrow {x^2} \cdot {x^2} + {x^2} \cdot 2x + {x^2} \cdot 5$
Simplifying we get,
$ \Rightarrow {x^4} + 2{x^3} + 5{x^2}$
Now we multiply the second term of $({x^2} + 2x - 1)$, which is $2x$ to each term of $({x^2} + 2x + 5)$, we get,
$ \Rightarrow 2x \cdot ({x^2} + 2x + 5)$
$ \Rightarrow 2x \cdot {x^2} + 2x \cdot 2x + 2x \cdot 5$
Simplifying we get,
$ \Rightarrow 2{x^3} + 4{x^2} + 10x$
Now we multiply the third term of $({x^2} + 2x - 1)$, which is $ - 1$ to each term of $({x^2} + 2x + 5)$, we get,
$ \Rightarrow ( - 1) \cdot ({x^2} + 2x + 5)$
$ \Rightarrow ( - 1) \cdot {x^2} + ( - 1) \cdot 2x + ( - 1) \cdot 5$
Simplifying we get,
$ \Rightarrow - {x^2} - 2x - 5$
Now we add all the expressions obtained above and simplify further.
Adding all the expressions, we get,
$ \Rightarrow {x^4} + 2{x^3} + 5{x^2} + 2{x^3} + 4{x^2} + 10x - {x^2} - 2x - 5$
Rearranging the terms we get,
$ \Rightarrow {x^4} + 2{x^3} + 2{x^3} + 5{x^2} + 4{x^2} - {x^2} + 10x - 2x - 5$
Combining like terms, $2{x^3} + 2{x^3} = 4{x^3}$
Combining like terms, $5{x^2} + 4{x^2} - {x^2} = 8{x^2}$
Combining like terms, $10x - 2x = 8x$
Hence we get,
$ \Rightarrow {x^4} + 4{x^3} + 8{x^2} + 8x - 5$

Thus multiplying the polynomial $({x^2} + 2x - 1)({x^2} + 2x + 5)$ we get the resultant polynomial as ${x^4} + 4{x^3} + 8{x^2} + 8x - 5$.

Note: Students must be careful while multiplying the polynomials. There may be a situation, where students miss the terms in the polynomial to multiply. That’s why we multiply term by term. So that there won’t be any confusion. And it will be easier to simplify the expression. The highest power of the polynomial is called the degree of the polynomial. After getting the final expression, we must arrange the terms from highest power to lowest power.