Question

Which of the following is not a factor of the polynomial $\mathbf{45}{a}^2-\mathbf{80}{b}^2$?

F 5

G$\mathbf{3}a-\mathbf{4}b$

H$\mathbf{2}a-\mathbf{5}b$

J$\mathbf{3}a+\mathbf{4}b$

Short answer

$\mathbf{2}a-\mathbf{5}b$is not a factor of $45a^2-80b^2$.

Step-by-step solution

Step 1. Factor each number.

Factor$45a^2-80b^2$ and write all powers of variables as products.

$$\begin{gathered} 45a^2-80b^2=5\left[9a^2-16b^2\right] \\ =5\left[{\left(3a\right)}^2-{\left(4b\right)}^2\right] \end{gathered}$$

Step 2. Write rule of product of a sum and a difference.

The product of$x+y$ and$x-y$ is the square of x minus the square of y, that is,

$\left(x+y\right)\left(x-y\right)=x^2-y^2$

Step 3. Apply rule of product of a sum and a difference.

Apply the rule of product of a sum and a difference to the obtained expression.

$5\left(3a+4b\right)\left(3a-4b\right)$

Step 4. Write the conclusion.

From the obtained answer, it can be observed that options F, G and J matches with the answer and hence,$2a-5b$ is not a factor of a given polynomial. Therefore, H the required option.