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Chapter 5: Problem 16

Find each product. $$ 4 x(5 x+3) $$

Short Answer

Expert verified
The product is \(20x^2 + 12x\).

Step by step solution

01

- Distribute the First Term

Multiply the first term inside the parentheses by 4x. This gives us: \(4x \times 5x = 20x^2\).
02

- Distribute the Second Term

Next, multiply the second term inside the parentheses by 4x. This gives us: \(4x \times 3 = 12x\).
03

- Combine the Results

Add the results from the previous steps together: \(20x^2 + 12x\).

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Key Concepts

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

Distribution
Understanding distribution is crucial in polynomial multiplication. Distribution in algebra involves multiplying a single term by each term inside a set of parentheses. This is often presented as:

\(a(b + c) = ab + ac\).

In our exercise, we have \(4x(5x + 3)\). We need to multiply \(4x\) by \(5x\) and then by \(3\), distributing \(4x\) to each term inside the parentheses.
  • First, \(4x \times 5x = 20x^2 \).
  • Second, \(4x \times 3 = 12x \).
Combining these results, we get \(20x^2 + 12x\). This is the basic concept of distribution.
Monomials
A monomial is an algebraic expression consisting of a single term. Monomials can include numbers, variables, or the product of both with non-negative integer exponents.

Examples of monomials include:
  • \(7y\)
  • \(-3a^3\)
  • \(5xyz\)
In our exercise, the expressions \(4x\), \(5x\), and \(3\) are all monomials. We multiply these monomials by following the rules of distribution.
Binomials
A binomial contains exactly two terms. These terms are separated by either a plus or minus sign. For example:
  • \(a + b\)
  • \(x^2 - y^2\)
  • \(3m + 4n\)
In the given exercise, \(5x + 3\) is a binomial because it consists of two terms: \(5x\) and \(3\). To solve the problem using distribution, we handle each term of the binomial separately.
Combining Like Terms
Combining like terms is the process of simplifying expressions by adding or subtracting terms that have identical variable parts. Like terms are terms that contain the same variables raised to the same power.

For example:
  • \(3x + 5x = 8x\) (since both terms have \(x\))
  • \(2y^2 - 4y^2 = -2y^2\) (since both terms have \(y^2\))
In our problem, after distributing, we end up with \(20x^2\) and \(12x\). There are no like terms to combine here, so the final expression stays as \(20x^2 + 12x\).

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Most popular questions from this chapter

Find each product. \((3 r+5)(2 r+1)\)

Write each product as a sum of terms. Write answers with positive exponents only. Simplify each term. See Section 1.8 $$ \frac{1}{4 y}\left(y^{4}+6 y^{2}+8\right) $$

Perform each division. $$ \frac{5 x^{3}+4 x^{2}+10 x+20}{5 x+5} $$

Use scientific notation to calculate the answer to each problem. In \(2007,9.63 \times 10^{9}\) dollars were spent to attend motion pictures in the United States. Domestic admissions (the total number of tickets sold) for that year totaled 1.4 billion. What was the average ticket price?

Evaluate each expression for \(x=3\) $$ 4 x^{3}-5 x^{2}+2 x-5 $$
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