Multiply The Polynomial By The Monomial Using The Distributive Property And/or The Product Rule Of Exponents: { (2x)\left(-3x^2 + 3x - 7\right)$}$

Introduction

In algebra, multiplying polynomials by monomials is a fundamental concept that helps us simplify complex expressions. The distributive property and the product rule of exponents are two essential tools that we can use to achieve this. In this article, we will explore how to multiply the polynomial ${-3x^2 + 3x - 7}$ by the monomial ${2x}$ using these two methods.

The Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a monomial by a polynomial by multiplying each term of the polynomial by the monomial. In other words, we can distribute the monomial to each term of the polynomial. The distributive property can be expressed as:

${a(b + c) = ab + ac}$

where ${a}$, ${b}$, and ${c}$ are algebraic expressions.

Applying the Distributive Property

To multiply the polynomial ${-3x^2 + 3x - 7}$ by the monomial ${2x}$, we can use the distributive property. We will multiply each term of the polynomial by the monomial ${2x}$.

${(2x)\left(-3x^2 + 3x - 7\right) = (2x)(-3x^2) + (2x)(3x) + (2x)(-7)}$

Using the distributive property, we can simplify each term:

${(2x)(-3x^2) = -6x^3}$ ${(2x)(3x) = 6x^2}$ ${(2x)(-7) = -14x}$

Now, we can combine the simplified terms to get the final result:

${-6x^3 + 6x^2 - 14x}$

The Product Rule of Exponents

The product rule of exponents is a fundamental concept in algebra that allows us to multiply two monomials with the same base by adding their exponents. In other words, if we have two monomials with the same base, we can multiply them by adding their exponents. The product rule of exponents can be expressed as:

${a^m \cdot a^n = a^{m+n}}$

where ${a}$ is the base and ${m}$ and ${n}$ are the exponents.

Applying the Product Rule of Exponents

To multiply the polynomial ${-3x^2 + 3x - 7}$ by the monomial ${2x}$, we can use the product rule of exponents. We will multiply each term of the polynomial by the monomial ${2x}$.

${(2x)\left(-3x^2 + 3x - 7\right) = -6x^3 + 6x^2 - 14x}$

Using the product rule of exponents, we can simplify each term:

${-6x^3 = -6x^{3+1} = -6x^4}$ ${6x^2 = 6x^{2+1} = 6x^3}$ ${-14x = -14x^{1+1} = -14x^2}$

Now, we can combine the simplified terms to get the final result:

${-6x^4 + 6x^3 - 14x^2}$

Conclusion

In this article, we have explored how to multiply the polynomial ${-3x^2 + 3x - 7}$ by the monomial ${2x}$ using the distributive property and the product rule of exponents. We have seen that both methods can be used to achieve the same result, but the product rule of exponents is a more efficient method when dealing with monomials with the same base.

Examples and Exercises

1. Multiply the polynomial ${2x^2 + 3x - 1}$ by the monomial ${4x}$.
2. Multiply the polynomial ${x^2 - 2x + 1}$ by the monomial ${3x}$.
3. Multiply the polynomial ${2x^3 - 3x^2 + 1}$ by the monomial ${x}$.

Solutions

1. ${(4x)(2x^2) + (4x)(3x) + (4x)(-1) = 8x^3 + 12x^2 - 4x}$
2. ${(3x)(x^2) + (3x)(-2x) + (3x)(1) = 3x^3 - 6x^2 + 3x}$
3. ${(x)(2x^3) + (x)(-3x^2) + (x)(1) = 2x^4 - 3x^3 + x}$

Final Thoughts

Introduction

In our previous article, we explored how to multiply polynomials by monomials using the distributive property and the product rule of exponents. In this article, we will answer some frequently asked questions about multiplying polynomials by monomials.

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to multiply a monomial by a polynomial by multiplying each term of the polynomial by the monomial.

Q: How do I apply the distributive property to multiply a polynomial by a monomial?

A: To apply the distributive property, you need to multiply each term of the polynomial by the monomial. For example, if you want to multiply the polynomial ${2x^2 + 3x - 1}$ by the monomial ${4x}$, you would multiply each term of the polynomial by the monomial:

${(4x)(2x^2) + (4x)(3x) + (4x)(-1) = 8x^3 + 12x^2 - 4x}$

Q: What is the product rule of exponents?

A: The product rule of exponents is a fundamental concept in algebra that allows us to multiply two monomials with the same base by adding their exponents.

Q: How do I apply the product rule of exponents to multiply a polynomial by a monomial?

A: To apply the product rule of exponents, you need to multiply each term of the polynomial by the monomial and then add the exponents. For example, if you want to multiply the polynomial ${2x^2 + 3x - 1}$ by the monomial ${4x}$, you would multiply each term of the polynomial by the monomial and then add the exponents:

${(4x)(2x^2) = 8x^{2+1} = 8x^3}$ ${(4x)(3x) = 12x^{1+1} = 12x^2}$ ${(4x)(-1) = -4x^{1+1} = -4x^2}$

Q: What are some common mistakes to avoid when multiplying polynomials by monomials?

A: Some common mistakes to avoid when multiplying polynomials by monomials include:

• Not distributing the monomial to each term of the polynomial
• Not adding the exponents when using the product rule of exponents
• Not simplifying the expression after multiplying the polynomial by the monomial

Q: How do I simplify an expression after multiplying a polynomial by a monomial?

A: To simplify an expression after multiplying a polynomial by a monomial, you need to combine like terms and simplify the expression. For example, if you want to simplify the expression ${8x^3 + 12x^2 - 4x}$, you would combine like terms and simplify the expression:

${8x^3 + 12x^2 - 4x = 8x^3 + 12x^2 - 4x}$

Q: What are some real-world applications of multiplying polynomials by monomials?

A: Some real-world applications of multiplying polynomials by monomials include:

• Calculating the area of a rectangle
• Calculating the volume of a cube
• Calculating the surface area of a sphere

Conclusion

In this article, we have answered some frequently asked questions about multiplying polynomials by monomials. We have covered topics such as the distributive property, the product rule of exponents, and common mistakes to avoid when multiplying polynomials by monomials. By understanding and applying these concepts, you can solve a wide range of problems in algebra and beyond.

Examples and Exercises

1. Multiply the polynomial ${2x^2 + 3x - 1}$ by the monomial ${4x}$.
2. Multiply the polynomial ${x^2 - 2x + 1}$ by the monomial ${3x}$.
3. Multiply the polynomial ${2x^3 - 3x^2 + 1}$ by the monomial ${x}$.

Solutions

1. ${(4x)(2x^2) + (4x)(3x) + (4x)(-1) = 8x^3 + 12x^2 - 4x}$
2. ${(3x)(x^2) + (3x)(-2x) + (3x)(1) = 3x^3 - 6x^2 + 3x}$
3. ${(x)(2x^3) + (x)(-3x^2) + (x)(1) = 2x^4 - 3x^3 + x}$

Final Thoughts

Multiplying polynomials by monomials is a fundamental concept in algebra that helps us simplify complex expressions. By understanding and applying these concepts, you can solve a wide range of problems in algebra and beyond.