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Chapter 5: Problem 60

Multiply and check. $$ \left(a^{2}+4\right)\left(5 a^{3}-3 a-1\right) $$

Short Answer

Expert verified
The result is \[5a^5 + 17a^3 - a^2 - 12a - 4\] .

Step by step solution

01

- Apply the Distributive Property

To multiply the two polynomials, distribute each term in the first polynomial by each term in the second polynomial. Essentially, multiply each term in \(a^2 + 4\) with each term in \(5a^3 - 3a - 1\).
02

- Multiply Each Term

Perform the multiplication for each pair of terms: \[ a^2 \times 5a^3 = 5a^5 \] \[ a^2 \times (-3a) = -3a^3 \] \[ a^2 \times (-1) = -a^2 \] \[ 4 \times 5a^3 = 20a^3 \] \[ 4 \times (-3a) = -12a \] \[ 4 \times (-1) = -4 \]
03

- Combine Like Terms

Sum the results from the multiplication to combine like terms: \[ 5a^5 + (-3a^3 + 20a^3) + (-a^2) + (-12a) + (-4) = 5a^5 + 17a^3 - a^2 - 12a - 4 \]
04

- Write the Final Expression

The result of the polynomial multiplication is: \[ 5a^5 + 17a^3 - a^2 - 12a - 4 \]

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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 crucial tool used frequently in algebra. It allows us to multiply a single term by each term within a parenthesis. In polynomial multiplication, it is applied to break down complex problems into simpler parts.

For instance, in the exercise \( (a^2 + 4)(5a^3 - 3a - 1) \), we use the distributive property to distribute \(a^2\) and \(4\) to each term in \(5a^3 - 3a - 1\).

This method ensures that every combination of terms is considered, making it easier to handle long and complicated expressions.
multiplying polynomials
Multiplying polynomials involves several steps. Each term in the first polynomial must be multiplied by each term in the second polynomial. This systematic approach ensures we don't miss any part of the multiplication.

In our exercise, we first calculate:
\[ a^2 \times 5a^3 = 5a^5 \] \[ a^2 \times (-3a) = -3a^3 \] \[ a^2 \times (-1) = -a^2 \]
and then:
\[ 4 \times 5a^3 = 20a^3 \] \[ 4 \times (-3a) = -12a \] \[ 4 \times (-1) = -4 \]

By carrying out these multiplications, you break down the problem into manageable pieces.
combining like terms
After multiplying polynomials, the next step is to combine like terms. Like terms are terms that have the same variables raised to the same powers. Combining them simplifies the polynomial into a more understandable form.

From the exercise, we've distributed and multiplied to get:
\[ 5a^5 \] \[ -3a^3 + 20a^3 \] \[ -a^2 \] \[ -12a \] \[ -4 \]
By combining the like terms \(-3a^3 + 20a^3\), we obtain \(17a^3\). The other terms do not have like terms to combine with, so they remain unchanged.

Thus, the final expression is \(5a^5 + 17a^3 - a^2 - 12a - 4\).
algebraic expressions
An algebraic expression is a combination of variables, numbers, and at least one arithmetic operation. In our polynomial multiplication problem, \(a^2 + 4\) and \(5a^3 - 3a - 1\) are both algebraic expressions.

These expressions can become quite complex, but by using algebraic principles like the distributive property and combining like terms, we can simplify and solve them effectively.

Understanding how to handle algebraic expressions is key to mastering algebra.

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Most popular questions from this chapter

Simplify. Write \(8^{-3} \cdot 32 \div 16^{2}\) as a power of 2

Simplify. $$ \left[\left(5^{-3}\right)^{2}\right]^{-1} $$

Simplify. Assume that no denominator is zero and that \(0^{0}\) is not considered. $$ \left(\frac{5 a^{7}}{2 b^{5} c}\right)^{0} $$

For each pair of functions fand \(g\), determine the domain of the sum, the difference, and the product of the two functions. $$ \begin{aligned} &f(x)=3 x^{2}\\\ &g(x)=\frac{1}{x-9} \end{aligned} $$

For each pair of functions \(f\) and \(g\), determine the domain of \(f / g\) $$ \begin{aligned} &f(x)=5+x\\\ &g(x)=6-2 x \end{aligned} $$
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