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Chapter 5: Problem 23

Find each product. $$ -4 r^{3}\left(-7 r^{2}+8 r-9\right) $$

Short Answer

Expert verified
The product is \(28r^5 - 32r^4 + 36r^3\).

Step by step solution

01

- Distribute the first term

Distribute \(-4r^3\) to each term inside the parenthesis: \(-7r^2, 8r, -9\).
02

- Multiply with the first term inside parenthesis

Multiply \(-4r^3\) with \(-7r^2\): \[-4r^3 \times -7r^2 = 28r^5\].
03

- Multiply with the second term inside parenthesis

Multiply \(-4r^3\) with \( 8r \): \[-4r^3 \times 8r = -32r^4\].
04

- Multiply with the third term inside parenthesis

Multiply \(-4r^3\) with \( -9 \): \(-4r^3 \times -9 = 36r^3\).
05

- Combine the results

Combine all the products obtained: \[28r^5 - 32r^4 + 36r^3\].

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Key Concepts

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

Polynomial Multiplication
Polynomial multiplication involves multiplying each term in one polynomial by each term in another polynomial. In our example, we need to distribute the term \(-4r^3\) to each term inside the parentheses. Here are the steps:

- Multiply \(-4r^3\) by \(-7r^2\)
- Multiply \(-4r^3\) by \(8r\)
- Multiply \(-4r^3\) by \(-9\)

Each multiplication should be handled carefully, particularly paying attention to the signs and exponents. After completing the multiplication, the polynomial terms must be combined correctly, ensuring that each result is accounted for.
Exponents
Understanding how to handle exponents is crucial in polynomial multiplication. When multiplying terms with the same base, you add their exponents. For example, when multiplying \(-4r^3\) by \(-7r^2\), we combine the exponents of \(r\): \((r^3 \times r^2 = r^{3+2} = r^5)\).

Here鈥檚 the multiplication done step by step:

- Multiply the coefficients: \(-4 \times -7 = 28\)
- Add the exponents for \(r\): \(3 + 2 = 5\)

Putting it all together, the result is \(28r^5\). Apply the same process for the remaining terms inside the parenthesis.
Combining Like Terms
After completing the polynomial multiplication, you'll end up with several terms. These terms need to be combined appropriately. In our example, the terms obtained are \(28r^5, -32r^4,\) and \(36r^3\).

Combining like terms involves:

- Checking if any of the terms have the same exponent
- Grouping these terms together

In this case, since each resulting term has a unique exponent, there are no like terms to combine. Thus, the expression remains:

\[28r^5 - 32r^4 + 36r^3\]

Ensuring that you carefully combine terms and apply the correct arithmetic can prevent mistakes in your final result.

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Most popular questions from this chapter

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 \(\mathbf{A} .8\) \(\mathbf{B} .16\) \(\mathbf{C} .-\frac{1}{16}\) \(\mathbf{D} .-8\) \(\mathbf{E} .-16\) \(\mathbf{F} . \frac{1}{16}\)

The special product $$ (x+y)(x-y)=x^{2}-y^{2} $$ $$ \text { can be used to perform some multiplication problems. Here are two examples.} $$ $$ \begin{aligned} 51 \times 49 &=(50+1)(50-1) \\ &=50^{2}-1^{2} \\ &=2500-1^{2} \\ &=2499 \end{aligned} \quad | \begin{aligned} 102 \times 98 &=(100+2)(100-2) \\ &=100^{2}-2^{2} \\ &=10,000-4 \\ &=9996 \end{aligned} $$ Once these patterns are recognized, multiplications of this type can be done mentally. Use this method to calculate each product mentally. $$ 20 \frac{1}{2} \times 19 \frac{1}{2} $$

Perform each division. $$ \frac{-4 x+3 x^{3}+2}{x-1} $$

Simplify by writing each expression wth positive exponents. Assume that all variables represent nonzero real numbers. $$ \frac{\left(4 a^{2} b^{3}\right)^{-2}\left(2 a b^{-1}\right)^{3}}{\left(a^{3} b\right)^{-4}} $$

Perform each subtraction. $$8-(-4)$$
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