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Chapter 5: Problem 30

Multiply. $$\left(x^{4}-3\right)(2 x+1)$$

Short Answer

Expert verified
2x^5 + x^4 - 6x - 3

Step by step solution

01

- Distribute the First Term

Multiply the first term of the polynomial \( x^4 \) by each term inside the parentheses: \[x^4 \(2x + 1\) = x^4 \cdot 2x + x^4 \cdot 1 \]
02

- Multiply the Second Term

Now multiply each term inside the parentheses by the second term of the first polynomial, which is \(-3\): \[-3 \(2x + 1\) = -3 \cdot 2x + (-3) \cdot 1 \]
03

- Combine Like Terms

Add the two results from steps 1 and 2: \[\ x^4 \cdot 2x + x^4 \cdot 1 + (-3 \cdot 2x) + (-3 \cdot 1) = 2x^5 + x^4 - 6x - 3\]

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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 allows us to multiply each term inside a set of parentheses by another term outside the parentheses. This is essential when working with polynomials.

For instance, in the exercise, we have to multiply \((x^4 - 3)\) by \((2x + 1)\). First, we distribute \(x^4\) to each term in \((2x + 1)\).
\[ x^4 (2x + 1) = x^4 * 2x + x^4 * 1 \]
Then, distribute \(-3\) to each term in \((2x + 1)\).
\[ ( - 3) (2x + 1) = -3 * 2x + (-3) * 1 \]
This process ensures that every term in the first polynomial is multiplied by every term in the second polynomial, setting up for further simplification.
combining like terms
After distributing, you will often get terms with the same variables and exponents. These are known as 'like terms'. Combining like terms simplifies the expression.

From the exercise, after distributing, we have a mix of terms:
\[ x^4 * 2x + x^4 * 1 - 3 * 2x + (-3) * 1 \]
When we combine like terms, we group these terms based on their variables and powers to simplify. This gives us:
\[ 2x^5 + x^4 鈥� 6x 鈥� 3 \]
Notice we only have one \(x^5\) and one \(x^4\) term, so these remain as they are. For the \(x\) terms, we can simply add or subtract the coefficients.
This step is crucial for clarity and accuracy in polynomial expressions.
exponent rules
When multiplying polynomials, exponent rules are very important. They help us understand how to handle the variables' exponents during multiplication.

Let's look at our example again:
\[\begin{aligned}x^4(2x) \ = x^4 * 2x & = 2x^{ (4+1)} \ \text { since the rule of exponent addition applies}\ \]
When we multiply like bases, we add the exponents: in our case, \(2x^{(4+1)} = 2x^{5}\).
Another rule is any number raised to the first power is just that number, as seen here:
\[ x^4 * 1 = x^4 \]
Mastering these rules allows you to handle polynomial multiplication accurately and effortlessly.

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Most popular questions from this chapter

Review factoring expressions and solving equations. $$ x+5=0 $$

Simplify. $$ \frac{\left(\frac{1}{2}\right)^{3}\left(\frac{2}{3}\right)^{4}}{\left(\frac{5}{6}\right)^{3}} $$

To prepare for Section \(5.2,\) review operations with integers (Sections \(1.5-1.7)\) Perform the indicated operations. $$ 12-(-4) $$

For each pair of functions \(f\) and \(g,\) determine the domain of \(f / g\) $$ \begin{aligned} &f(x)=x^{4}\\\ &g(x)=x-3 \end{aligned} $$

For each pair of functions fand \(g\), determine the domain of the sum, the difference, and the product of the two functions. $$ \begin{aligned} &f(x)=\frac{1}{x-3}\\\ &g(x)=4 x^{3} \end{aligned} $$
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