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Chapter 5: Problem 44

Find each product of the monomial and the polynomial. $$5 x^{2}(x+6)$$

Short Answer

Expert verified
The product of the monomial \(5x^{2}\) and the binomial \((x+6)\) is \(5x^{3} + 30x^{2}\)

Step by step solution

01

Distribute the Monomial to Each Term of the Binomial

By using the distributive property, multiply the monomial \(5x^{2}\) by each term in the binomial \((x+6)\), which gives: \(5x^{2} * x + 5x^{2} * 6\)
02

Carry Out the Multiplication

Next, simplify the multiplication by applying the laws of exponents and carrying out the multiplication operation: \(5x^{3}+ 30x^{2}\)
03

Write the Final Answer

The final answer after simplifying is: \(5x^{3}+ 30x^{2}\)

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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 helps simplify expressions and solve equations. It allows us to multiply a single term across multiple terms within parentheses. This property is especially important when dealing with multiplication involving polynomials.
To use the distributive property, follow these simple steps:
  • Identify the term outside the parentheses (the monomial) and the terms inside the parentheses (usually a binomial or polynomial).
  • Multiply the monomial by each term inside the parentheses separately.
  • Add the results together to get the complete expression.
In our exercise, the monomial is \(5x^2\) and the binomial is \((x + 6)\). Applying the distributive property, we multiply \(5x^2\) by each term inside the parentheses, resulting in \(5x^{2} \cdot x\) and \(5x^{2} \cdot 6\). This operation simplifies the expression by breaking it down to more manageable parts.
Monomials
A monomial is a mathematical expression that consists of a single term. It can include numbers, variables, and positive integer exponents of variables, such as \(5x^2\). Monomials are the building blocks in algebraic expressions and polynomial equations.
Characteristics of monomials include:
  • Contain no addition or subtraction. They consist of one term only.
  • May contain variables raised to whole number exponents.
  • Include a numerical coefficient, which is the number multiplied by the variable(s).
In the exercise, \(5x^2\) is a monomial. It is composed of a coefficient (5) and a variable \(x\) raised to the power of 2. Understanding monomials allows us to apply operations like multiplication and addition more effectively in algebra.
Exponents
Exponents are used in mathematics to denote repeated multiplication of a number by itself. They are a compact way of expressing such operations and come into play strongly in algebra.
Some basics of exponents are:
  • The base is the number being multiplied.
  • The exponent, or power, tells us how many times the base is multiplied by itself.
  • \(x^n\) means \(x\) is multiplied by itself \(n\) times.
In the step-by-step solution, applying the laws of exponents was crucial for simplifying \(5x^{2} \cdot x\), resulting in \(5x^{3}\) since according to the rule \(x^a \cdot x^b = x^{a+b}\). Understanding exponents helps in multiplying terms efficiently and arriving at simplified algebraic expressions.
Binomials
A binomial is a type of polynomial with exactly two terms. Binomials are simple yet powerful components used across many mathematical operations, including addition, subtraction, and multiplication.
Some key points about binomials:
  • They consist of two unlike terms joined by an addition or subtraction sign.
  • Examples include \(x + 6\) or \(3a - b\).
  • Each term in a binomial can itself be a monomial.
In our exercise, the binomial is \((x + 6)\). When the distributive property is applied, each term in the binomial is separately multiplied by the monomial preceding it. This helps to expand and simplify expressions, enabling us to solve more complex algebraic problems with ease.

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Most popular questions from this chapter

Simplify: \((-10)(-7) \div(1-8)\)

Use a graphing utility to determine whether the divisions have been performed correctly. Graph each side of the given equation in the same viewing rectangle. The graphs should coincide. If they do not, correct the expression on the right side by using polynomial division. Then use your graphing utility to show that the division has been performed correctly. $$\frac{2 x^{2}+13 x+15}{x-5}=2 x+3$$

$$\text { Graph: } y=\frac{1}{3} x+2$$

Will help you prepare for the material covered in the next section. a. Find the missing exponent, designated by the question mark, in each final step. $$\begin{array}{l}\frac{7^{3}}{7^{5}}=\frac{7 \cdot 7 \cdot 7}{7 \cdot 7 \cdot 7 \cdot 7 \cdot 7}=\frac{1}{7^{2}} \\ \frac{7^{3}}{7^{5}}=7^{3-5}=7^{?}\end{array}$$ b. Based on your two results for \(\frac{7^{3}}{7^{5}},\) what can you conclude?

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. Each statement applies to the division problem $$\frac{x^{3}+1}{x+1}$$ The purpose of writing \(x^{3}+1\) as \(x^{3}+0 x^{2}+0 x+1\) is to keep all like terms aligned.
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