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Chapter 5: Problem 49

Find each product of the monomial and the polynomial. $$\left(3 x^{3}+4 x^{2}\right)(2 x)$$

Short Answer

Expert verified
The product of the monomial and the polynomial \(\left(3 x^{3} + 4 x^{2}\right)(2 x)\) is \(6x^{4} + 8x^{3}\).

Step by step solution

01

- Identify the monomial and the polynomial

In the given exercise, \(2x\) is the monomial and \(3 x^{3} + 4 x^{2}\) is the polynomial.
02

- Distribute the monomial to each term of the polynomial

Now, applying the distributive law or the FOIL method, multiply the monomial \(2x\) with each term of the polynomial \(3x^{3} + 4x^{2}\) individually. Therefore, we get \(2x * 3x^{3} + 2x * 4x^{2}\).
03

- Simplify each term

After multiplication, simplify each term. Therefore, the product becomes \(6x^{4} + 8x^{3}\).

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Key Concepts

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

Understanding Monomials
A monomial is a term that consists of a single number or a product of numbers and variables raised to a power. It's the simplest form of polynomial expression. Monomials do not include addition or subtraction, just a single term. For instance, in our exercise, the expression \(2x\) is a monomial. It contains the constant 2 and the variable \(x\), which is raised to the power of 1. Here are some key points about monomials:
  • They can include coefficients (like the number 2 in \(2x\)).
  • They can include variables (such as \(x\)).
  • Exponents in monomials must be non-negative integers.
Monomials are the building blocks of more complex polynomial expressions. Their simplicity makes them very useful in algebra, especially when multiplying with polynomials.
Exploring Polynomials
A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Unlike monomials, polynomials can contain multiple terms. In the given exercise, \(3x^3 + 4x^2\) is the polynomial. This expression is made up of two terms. Let's break down some characteristics of polynomials.
  • Each term is a monomial.
  • The standard form of a polynomial arranges its terms in descending order of exponents.
  • Polynomials are a broader category that includes monomials, binomials, and trinomials based on the number of terms.
Polynomials are versatile and can represent complex relationships in variables. They are used in various fields, such as physics, engineering, and economics, due to their ability to model different scenarios.
The Distributive Law in Action
The distributive law is a fundamental property of numbers that applies to algebraic expressions. This law states that multiplying a single term by a sum of terms equates to multiplying each addend by the single term, and then summing the results. In our exercise, we apply the distributive law to multiply the monomial \(2x\) with each term of the polynomial \(3x^3 + 4x^2\). Here's how it works:
  • First, multiply \(2x\) by \(3x^3\), resulting in \(6x^4\).
  • Next, multiply \(2x\) by \(4x^2\), resulting in \(8x^3\).
  • Finally, add these results together to get \(6x^4 + 8x^3\).
The distributive law is essential for simplifying expressions and is a powerful tool in algebra. By understanding and applying it, you can efficiently solve polynomial multiplication and other algebraic operations.

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Most popular questions from this chapter

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. $$\frac{x^{3}+3 x^{2}+5 x+3}{x+1}=x^{2}-2 x+3$$

determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(\left(2 x^{2}-8 x+6\right)-\left(x^{2}-3 x+5\right)=x^{2}-5 x+1\) for any value of \(x .\)

perform the indicated operations. $$\begin{aligned} &\left[\left(7 y^{2}-4 y+2\right)-\left(12 y^{2}+3 y-5\right)\right]-\left[\left(5 y^{2}-2 y-8\right)+\left(-7 y^{2}+10 y-13\right)\right] \end{aligned}$$

determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The number of people who catch a cold \(t\) weeks after January 1 is \(5 t-3 t^{2}+t^{3}\) The number of people who recover \(t\) weeks after January 1 is \(t-t^{2}+\frac{1}{3} t^{3}\) Write a polynomial in standard form for the number of people who are still ill with a cold \(t\) weeks after January 1

In your own words, explain how to divide a polynomial by a binomial. Use \(\frac{x^{2}+4}{x+2}\) in your explanation.
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