Multiply The Polynomials Using The Distributive Property And Combine Like Terms.$\left(6x^2 - 4\right)(-x - 1$\]Answer: $\square$

Introduction

In algebra, multiplying polynomials is a fundamental operation that involves multiplying each term of one polynomial by each term of another polynomial. The distributive property is a key concept in multiplying polynomials, which states that for any real numbers a, b, and c, the following equation holds: a(b + c) = ab + ac. In this article, we will explore how to multiply polynomials using the distributive property and combine like terms.

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a single term by a sum of terms. The distributive property can be written as:

a(b + c) = ab + ac

This means that we can multiply a single term, a, by a sum of terms, b + c, and get the same result as multiplying a by each term separately and then adding the results.

Multiplying Polynomials Using the Distributive Property

To multiply polynomials using the distributive property, we need to multiply each term of one polynomial by each term of another polynomial. Let's consider the example given in the problem:

$\left(6x^2 - 4\right)(-x - 1)$

To multiply these two polynomials, we need to multiply each term of the first polynomial by each term of the second polynomial.

Step 1: Multiply the First Term of the First Polynomial by Each Term of the Second Polynomial

The first term of the first polynomial is 6x^2. We need to multiply this term by each term of the second polynomial, which are -x and -1.

6x^2(-x) = -6x^3 6x^2(-1) = -6x^2

Step 2: Multiply the Second Term of the First Polynomial by Each Term of the Second Polynomial

The second term of the first polynomial is -4. We need to multiply this term by each term of the second polynomial, which are -x and -1.

-4(-x) = 4x -4(-1) = 4

Step 3: Combine Like Terms

Now that we have multiplied each term of the first polynomial by each term of the second polynomial, we need to combine like terms. Like terms are terms that have the same variable and exponent.

-6x^3 is a term with a variable of x and an exponent of 3. -6x^2 is a term with a variable of x and an exponent of 2. 4x is a term with a variable of x and an exponent of 1. 4 is a constant term.

We can combine the like terms by adding or subtracting the coefficients of the like terms.

-6x^3 is a term with a coefficient of -6. -6x^2 is a term with a coefficient of -6. 4x is a term with a coefficient of 4. 4 is a constant term with a coefficient of 4.

We can combine the like terms as follows:

-6x^3 + (-6x^2) + 4x + 4

Combining the like terms, we get:

-6x^3 - 6x^2 + 4x + 4

Conclusion

In this article, we have explored how to multiply polynomials using the distributive property and combine like terms. We have used the example given in the problem to demonstrate how to multiply polynomials using the distributive property and combine like terms. By following the steps outlined in this article, you should be able to multiply polynomials using the distributive property and combine like terms.

Example Problems

Here are some example problems that you can try to practice multiplying polynomials using the distributive property and combining like terms.

Example Problem 1

$\left(2x^2 + 3\right)(x - 2)$

Example Problem 2

$\left(x^2 - 4\right)(x + 2)$

Example Problem 3

$\left(3x^2 + 2\right)(x - 1)$

Answer Key

Here are the answers to the example problems.

Example Problem 1

$\left(2x^2 + 3\right)(x - 2)$

= 2x^2(x) + 2x^2(-2) + 3(x) + 3(-2) = 2x^3 - 4x^2 + 3x - 6

Example Problem 2

$\left(x^2 - 4\right)(x + 2)$

= x^2(x) + x^2(2) - 4(x) - 4(2) = x^3 + 2x^2 - 4x - 8

Example Problem 3

$\left(3x^2 + 2\right)(x - 1)$

Introduction

In our previous article, we explored how to multiply polynomials using the distributive property and combine like terms. In this article, we will answer some frequently asked questions about multiplying polynomials using the distributive property and combining like terms.

Q&A

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to multiply a single term by a sum of terms. The distributive property can be written as:

a(b + c) = ab + ac

This means that we can multiply a single term, a, by a sum of terms, b + c, and get the same result as multiplying a by each term separately and then adding the results.

Q: How do I multiply polynomials using the distributive property?

A: To multiply polynomials using the distributive property, you need to multiply each term of one polynomial by each term of another polynomial. Let's consider the example given in the problem:

$\left(6x^2 - 4\right)(-x - 1)$

To multiply these two polynomials, you need to multiply each term of the first polynomial by each term of the second polynomial.

Q: What are like terms?

A: Like terms are terms that have the same variable and exponent. For example, 2x and 4x are like terms because they have the same variable (x) and exponent (1). Similarly, 3x^2 and 2x^2 are like terms because they have the same variable (x) and exponent (2).

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the like terms. For example, if you have the terms 2x and 4x, you can combine them by adding their coefficients:

2x + 4x = 6x

Similarly, if you have the terms 3x^2 and 2x^2, you can combine them by adding their coefficients:

3x^2 + 2x^2 = 5x^2

Q: What are some common mistakes to avoid when multiplying polynomials using the distributive property?

A: Some common mistakes to avoid when multiplying polynomials using the distributive property include:

• Forgetting to multiply each term of one polynomial by each term of another polynomial.
• Not combining like terms correctly.
• Making errors when multiplying or adding coefficients.

Q: How can I practice multiplying polynomials using the distributive property and combining like terms?

A: You can practice multiplying polynomials using the distributive property and combining like terms by working through example problems. You can also try using online resources or math software to help you practice.

Example Problems

Here are some example problems that you can try to practice multiplying polynomials using the distributive property and combining like terms.

Example Problem 1

$\left(2x^2 + 3\right)(x - 2)$

Example Problem 2

$\left(x^2 - 4\right)(x + 2)$

Example Problem 3

$\left(3x^2 + 2\right)(x - 1)$

Answer Key

Here are the answers to the example problems.

Example Problem 1

$\left(2x^2 + 3\right)(x - 2)$

= 2x^2(x) + 2x^2(-2) + 3(x) + 3(-2) = 2x^3 - 4x^2 + 3x - 6

Example Problem 2

$\left(x^2 - 4\right)(x + 2)$

= x^2(x) + x^2(2) - 4(x) - 4(2) = x^3 + 2x^2 - 4x - 8

Example Problem 3

$\left(3x^2 + 2\right)(x - 1)$

= 3x^2(x) + 3x^2(-1) + 2(x) + 2(-1) = 3x^3 - 3x^2 + 2x - 2

Conclusion

In this article, we have answered some frequently asked questions about multiplying polynomials using the distributive property and combining like terms. We have also provided example problems and answers to help you practice multiplying polynomials using the distributive property and combining like terms. By following the steps outlined in this article, you should be able to multiply polynomials using the distributive property and combine like terms with confidence.