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Chapter 4: Problem 34

Find each product. $$ (9 a+2)\left(9 a^{2}+a+1\right) $$

Short Answer

Expert verified
81a^3 + 27a^2 + 11a + 2

Step by step solution

01

Apply the distributive property

To find the product (9a + 2)(9a^2 + a + 1), apply the distributive property by multiplying each term in the first polynomial by each term in the second polynomial.
02

Multiply 9a by each term in the second polynomial

First, multiply 9a by 9a^2, a, and 1 respectively: 9a * 9a^2 = 81a^3 9a * a = 9a^2 9a * 1 = 9a
03

Multiply 2 by each term in the second polynomial

Next, multiply 2 by 9a^2, a, and 1 respectively: 2 * 9a^2 = 18a^2 2 * a = 2a 2 * 1 = 2
04

Add all the products together

Now add all the products obtained from steps 2 and 3: (81a^3 + 9a^2 + 9a) + (18a^2 + 2a + 2) Combine like terms: = 81a^3 + (9a^2 + 18a^2) + (9a + 2a) + 2 = 81a^3 + 27a^2 + 11a + 2

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

distributive property
The distributive property is crucial in polynomial multiplication. It says that you multiply each term inside a parenthesis by the term outside it. In simple terms, distribute the multiplication over addition or subtraction. For example, in the expression \((9a + 2)(9a^2 + a + 1)\), you distribute each term in \(9a + 2\) to each term in \(9a^2 + a + 1\).

Here鈥檚 how it looks step by step:
  • Multiply \((9a)\) by \(9a^2\), \(a\), and \(1\).
  • Multiply \((2)\) by \(9a^2\), \(a\), and \(1\).
After doing this, you get all the individual products, which you can then add together.
combining like terms
Combining like terms is your next step after distributing. Like terms are terms that have the same variables raised to the same power. In your result from the distributive property, you'll have several terms that need to be combined:
  • Group all \(a^3\) terms together.
  • Group all \(a^2\) terms together.
  • Group all \(a\) terms together.

For instance, after expanding \((9a + 2)(9a^2 + a + 1)\), you'll get: \[81a^3 + 9a^2 + 9a + 18a^2 + 2a + 2\]
Combine like terms:
  • Combine \((9a^2 + 18a^2) = 27a^2\).
  • Combine \((9a + 2a) = 11a\).
Now the expression looks simpler!
algebraic expressions
Understanding algebraic expressions is key to solving polynomial multiplication. An algebraic expression is a mix of variables, constants, and arithmetic operations (like addition, subtraction, etc.). In this case, we have \((9a + 2)\) and \((9a^2 + a + 1)\).

Here are some important points:
  • Each term in an algebraic expression is separated by a plus or minus sign.
  • The degree of a term is the power of the variable.
  • Polynomial expressions can have different degrees, but they can often be simplified.
Knowing this helps you properly distribute and combine terms. It also makes complex problems seem easier.

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Most popular questions from this chapter

Perform each indicated operation. \(\left(x^{2}+x\right)-\left(3 x^{2}+2 x-1\right)\)

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ -2 x^{5}\left(3 x^{2}+2 x-5\right)(4 x+2) $$

Add or subtract as indicated. \(\left(2 c^{4} d+3 c^{2} d^{2}-4 d^{2}\right)-\left(c^{4} d+8 c^{2} d^{2}-5 d^{2}\right)\)

Use scientific notation to calculate the result in each expression. Write answers in scientific notation. 75\. \(\frac{0.00000072(0.00023)}{0.000000018}\)

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ (x+7)^{2} $$
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