Question

How do you multiply $\left( 5x-1 \right)\left( 5x+1 \right)?$

Answer 1

Hint: We will use the polynomial multiplication. We will multiply each term of the first polynomial with each term of the second polynomial. Also, remember the identity $a{{x}^{n}}\cdot b{{x}^{m}}=a\cdot b\cdot {{x}^{n+m}}.$

Complete step-by-step solution:
Let us consider the given polynomial $\left( 5x-1 \right)\left( 5x+1 \right).$
Here, we are asked to multiply two polynomials of degree $1.$
The polynomials given are $5x-1$ and $5x+1.$
We are using the polynomial multiplication to find the product of the above two polynomials of degree $1.$
We know the basic rules of the polynomial multiplication.
In polynomial multiplication, we multiply terms without considering the exponents.
That is, ${{x}^{m}}\cdot {{x}^{n}}={{x}^{m+n}}.$
Also $a{{x}^{n}}\cdot b{{x}^{m}}=a\cdot b\cdot {{x}^{n+m}}.$
We consider $1$ as ${{x}^{0}}.$
Therefore, $a$ can be equated to $a{{x}^{0}}$ where $a$ is any constant and is equal to $a\cdot 1.$
So, here, in our problem we multiply $5x$ from the first polynomial $5x-1$ and $5x$ from the second polynomial $5x+1.$ Then we multiply $-1$ from the first polynomial with $5x$ from the second polynomial. We multiply $5x$ from the first polynomial with $1$ from the second polynomial. Finally, we multiply $-1$ from the first polynomial with $1$ from the second polynomial.
When we multiply $5x$ from the first polynomial with $5x$ from the second polynomial, we get $5x\cdot 5x=5\cdot 5\cdot x\cdot x.$
We can see that $5x\cdot 5x=25{{x}^{2}}.$
Now we consider the first term $5x$ of the first polynomial and the second term $1$ of the second polynomial.
We get, $5x\cdot 1=5x.$
In the next step, we consider the second term of the first polynomial and the first term of the second polynomial, we get $-1\cdot 5x=-5x.$
Now the second term of the two polynomials are to be multiplied, we get $-1\cdot 1=-1.$
So, the product is $\left( 5x-1 \right)\left( 5x+1 \right)=25{{x}^{2}}-5x+5x-1=25{{x}^{2}}-1.$
Hence, $\left( 5x-1 \right)\left( 5x+1 \right)=25{{x}^{2}}-1.$

Note: Remember the identity $\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}.$ Let us apply the identity to get the result directly. The product of the polynomial $\left( 5x-1 \right)\left( 5x+1 \right)={{\left( 5x \right)}^{2}}-{{1}^{2}}=25{{x}^{2}}-1.$