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Chapter 5: Problem 59

Find each product. $$\left(3 a^{2} b+a\right)\left(3 a^{2} b-a\right)$$

Short Answer

Expert verified
The product of the binomials \(3a^2b + a\) and \(3a^2b - a\) is \(9a^4b^2 - a^2\).

Step by step solution

01

Expand the product rule

Expanding \( (3a^2b + a) \) and \( (3a^2b - a) \) using the distributive law gives us: \(3a^2b(3a^2b) - a(3a^2b) + a(3a^2b) - a(a)\)
02

Simplify the terms

Next, simplify these terms using the rules of indices. \(3a^2b(3a^2b)\) simplifies to \(9a^4b^2\), \(a(3a^2b)\) simplifies to \(3a^3b\), and \(a(a)\) simplifies to \(a^2\)
03

Combine like terms

Additionally, observe that the terms \( - a(3a^2b) \) and \( + a(3a^2b) \) are of similar kind, and hence they can be added together to become \(0\). So, the equation \(9a^4b^2 - a^2\) remains.
04

Write the final result

The final result of the multiplication of the two binomials \(3a^2b + a\) and \(3a^2b - a\) is \(9a^4b^2 - a^2\).

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Most popular questions from this chapter

The mad Dr. Frankenstein has gathered enough bits and pieces (so to speak) for \(2^{-1}+2^{-2}\) of his creature-to-be. Write a fraction that represents the amount of his creature that must still be obtained.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$5^{-2}>2^{-5}$$

Use a vertical format to find each product. $$\begin{array}{l}x^{2}+7 x-3 \\\x^{2}-x-1 \\\\\hline\end{array}$$

Will help you prepare for the material covered in the next section. In each exercise, find the indicated products. Then, if possible, state a fast method for finding these products. (You may already be familiar with some of these methods from a high school algebra course.) a. \((x+3)(x+4)\) b. \((x+5)(x+20)\)

Use a vertical format to find each product. $$\begin{array}{r}9 y^{3}-7 y^{2}+5 y \\\3 y^{2}+5 y \\\\\hline\end{array}$$
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