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Chapter 5: Problem 24

Find each product. $$ -9 a^{5}\left(-3 a^{6}-2 a^{4}+8 a^{2}\right) $$

Short Answer

Expert verified
27a^{11} + 18a^9 - 72a^7

Step by step solution

01

Distribute \(-9a^5\)

Multiply \(-9a^5\) by each term inside the parentheses \(-3a^6\), \(-2a^4\), and \(+8a^2\).
02

Multiply \(-9a^5\) by \(-3a^6\)

The product of \(-9a^5 \) and \(-3a^6\) is: \[ (-9a^5) \times (-3a^6) = 27a^{11} \]
03

Multiply \(-9a^5\) by \(-2a^4\)

The product of \(-9a^5\) and \(-2a^4\) is: \(( -9a^5 ) \times (-2a^4) = 18a^9\)
04

Multiply \(-9a^5\) by \[8a^2\]

The product of \(-9a^5\) and \(+8a^2\) is: \(( -9a^5 ) \times 8a^2 = -72a^7\)
05

Combine the results

Add all the products together: \[ 27a^{11} + 18a^9 - 72a^7 \]

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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 a fundamental concept in algebra that allows you to multiply a single term by each term within a parenthesis. In our exercise, we start with the expression \(-9a^5(-3a^6 - 2a^4 + 8a^2)\). According to the distributive property, we need to distribute \(-9a^5\) to each term inside the parenthesis. This means we'll multiply \(-9a^5\) by \(-3a^6\), \(-2a^4\), and \(8a^2\).

Here's a breakdown of that process:
  • Multiply \((-9a^5)\) by \((-3a^6)\): This gives \(27a^{11}\)
  • Multiply \((-9a^5)\) by \((-2a^4)\): This results in \(18a^9\)
  • Multiply \((-9a^5)\) by \((8a^2)\): We get \(-72a^7\)

exponent rules
Understanding exponent rules is crucial when performing polynomial multiplication. These rules help simplify expressions involving powers of variables. In our exercise, we deal with multiplying terms like \(-9a^5\) and \(-3a^6\). When we multiply these, we apply the exponent rule that states:
\[ a^m \times a^n = a^{m+n} \]
This means that we add the exponents together. So, multiplying \(-9a^5\) by \(-3a^6\) gives us: \((-9) \times (-3) = 27\) and \((a^5 \times a^6 = a^{11})\). Together, this is \(27a^{11}\).

We follow the same rule for multiplying the other terms:
  • \((-9a^5) \times (-2a^4) = 18a^9\)
  • \((-9a^5) \times 8a^2 = -72a^7\)

These steps ensure that we're accurately multiplying the coefficients and adding the exponents.
combining like terms
After distributing and using the exponent rules to simplify the expression, we are left with several terms. The next step involves combining like terms. Like terms are those that contain the same variables raised to the same power.

In our example, we end up with \(27a^{11} + 18a^9 - 72a^7\). Because these terms have different exponents (11, 9, and 7), they are not like terms. Therefore, they can't be combined further.

When solving more complex polynomial expressions, remember to:
  • Add or subtract coefficients of like terms
  • Keep the variable and its exponent unchanged
  • Simplify the expression as much as possible
These steps help to organize and simplify polynomial expressions effectively.

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Most popular questions from this chapter

Perform each division. $$ \frac{y^{3}-1}{y-1} $$

Match each expression in Column I with the equivalent expression in Column II. Choices in Column II may be used once, more than once, or not at all. An Exercise \(17, x \neq 0 .\). I (a) \(x^{0}\) (b) \(-x^{0}\) (c) \(7 x^{0}\) (d) \((7 x)^{0}\) (e) \(-7 x^{0}\) (f) \((-7 x)^{0}\) II A. 0 B. 1 C. -1 D. 7 E. -7 F. \(\frac{1}{7}\)

Match each expression in Column I with the equivalent expression in Column II. Choices in Column II may be used once, more than once, or not at all. An Exercise \(17, x \neq 0 .\). I (a) \(-2^{-4}\) (b) \((-2)^{-4}\) (c) \(2^{-4}\) (d) \(\frac{1}{2^{-4}}\) (e) \(\frac{1}{-2^{-4}}\) (f) \(\frac{1}{(-2)^{-4}}\) II \(\mathbf{A} .8\) \(\mathbf{B} .16\) \(\mathbf{C} .-\frac{1}{16}\) \(\mathbf{D} .-8\) \(\mathbf{E} .-16\) \(\mathbf{F} . \frac{1}{16}\)

Perform each division. $$ \frac{x^{4}-4 x^{3}+5 x^{2}-3 x+2}{x^{2}+3} $$

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