Expanding the Product of Two Binomials

Oct 13, 2025 by magicalceylontours.com 106 views

Introduction

In algebra, expanding and simplifying expressions is a crucial skill that helps you solve equations and inequalities. One of the most common types of expressions you'll encounter is the product of two or more polynomials. In this article, we'll focus on expanding and simplifying the product of two binomials, specifically the expression $(x+2)(x^2-2x+4)$ . By the end of this article, you'll be able to simplify this expression and understand the steps involved in expanding and simplifying algebraic expressions.

What is Expanding and Simplifying Algebraic Expressions?

Expanding and simplifying algebraic expressions involves multiplying and combining like terms to simplify an expression. This process is essential in solving equations and inequalities, as it helps you isolate the variable and find the solution. When expanding and simplifying expressions, you'll often encounter products of two or more polynomials, which can be simplified using the distributive property.

The Distributive Property

The distributive property is a fundamental concept in algebra that allows you to expand and simplify expressions. It states that for any real numbers $a$ , $b$ , and $c$ , the following equation holds:

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

This property can be applied to any polynomial expression, including the product of two binomials.

Expanding the Product of Two Binomials

To expand the product of two binomials, you'll use the distributive property to multiply each term in the first binomial by each term in the second binomial. Let's apply this process to the expression $(x+2)(x^2-2x+4)$ .

Step 1: Multiply the First Term in the First Binomial by Each Term in the Second Binomial

The first term in the first binomial is $x$ , and the second term is $2$ . Multiply each term in the second binomial by $x$ :

$x(x^2) = x^3$ $x(-2x) = -2x^2$ $x(4) = 4x$

Step 2: Multiply the Second Term in the First Binomial by Each Term in the Second Binomial

The second term in the first binomial is $2$ . Multiply each term in the second binomial by $2$ :

$2(x^2) = 2x^2$ $2(-2x) = -4x$ $2(4) = 8$

Step 3: Combine Like Terms

Now that you've multiplied each term in the first binomial by each term in the second binomial, combine like terms to simplify the expression:

$x^3 - 2x^2 + 4x + 2x^2 - 4x + 8$

Combine the like terms:

$x^3 + 2x^2 + 8$

Simplifying the Expression

The expression $(x+2)(x^2-2x+4)$ can be simplified to $x^3 + 2x^2 + 8$ . This is the final answer.

Conclusion

Expanding and simplifying algebraic expressions is a crucial skill in mathematics. By understanding the distributive property and applying it to the product of two binomials, you can simplify expressions like $(x+2)(x^2-2x+4)$ . Remember to multiply each term in the first binomial by each term in the second binomial and combine like terms to simplify the expression. With practice, you'll become proficient in expanding and simplifying algebraic expressions, and you'll be able to solve equations and inequalities with confidence.

Frequently Asked Questions

Q: What is the distributive property? A: The distributive property is a fundamental concept in algebra that allows you to expand and simplify expressions. It states that for any real numbers $a$ , $b$ , and $c$ , the following equation holds: $a(b+c) = ab + ac$ .
Q: How do I expand the product of two binomials? A: To expand the product of two binomials, multiply each term in the first binomial by each term in the second binomial and combine like terms.
Q: What is the final answer to the expression $(x+2)(x^2-2x+4)$ ? A: The final answer to the expression $(x+2)(x^2-2x+4)$ is $x^3 + 2x^2 + 8$ .

Additional Resources

Khan Academy: Expanding and Simplifying Algebraic Expressions
Mathway: Expanding and Simplifying Algebraic Expressions
Wolfram Alpha: Expanding and Simplifying Algebraic Expressions

Final Answer

The final answer to the expression $(x+2)(x^2-2x+4)$ is $x^3 + 2x^2 + 8$ .

Introduction

What is Expanding and Simplifying Algebraic Expressions?

The Distributive Property

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

This property can be applied to any polynomial expression, including the product of two binomials.

Expanding the Product of Two Binomials

Step 1: Multiply the First Term in the First Binomial by Each Term in the Second Binomial

The first term in the first binomial is $x$ , and the second term is $2$ . Multiply each term in the second binomial by $x$ :

$x(x^2) = x^3$ $x(-2x) = -2x^2$ $x(4) = 4x$

Step 2: Multiply the Second Term in the First Binomial by Each Term in the Second Binomial

The second term in the first binomial is $2$ . Multiply each term in the second binomial by $2$ :

$2(x^2) = 2x^2$ $2(-2x) = -4x$ $2(4) = 8$

Step 3: Combine Like Terms

Now that you've multiplied each term in the first binomial by each term in the second binomial, combine like terms to simplify the expression:

$x^3 - 2x^2 + 4x + 2x^2 - 4x + 8$

Combine the like terms:

$x^3 + 2x^2 + 8$

Simplifying the Expression

The expression $(x+2)(x^2-2x+4)$ can be simplified to $x^3 + 2x^2 + 8$ . This is the final answer.

Conclusion

Frequently Asked Questions

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows you to expand and simplify expressions. It states that for any real numbers $a$ , $b$ , and $c$ , the following equation holds: $a(b+c) = ab + ac$ .

Q: How do I expand the product of two binomials?

A: To expand the product of two binomials, multiply each term in the first binomial by each term in the second binomial and combine like terms.

Q: What is the final answer to the expression $(x+2)(x^2-2x+4)$ ?

A: The final answer to the expression $(x+2)(x^2-2x+4)$ is $x^3 + 2x^2 + 8$ .

Q: Can I use the distributive property to simplify expressions with more than two binomials?

A: Yes, you can use the distributive property to simplify expressions with more than two binomials. However, you'll need to multiply each term in the first binomial by each term in the second binomial, and then multiply the result by each term in the third binomial, and so on.

Q: How do I know which terms to combine when simplifying an expression?

A: When simplifying an expression, combine like terms by adding or subtracting the coefficients of the same variable.

Q: Can I use a calculator to simplify expressions?

A: Yes, you can use a calculator to simplify expressions. However, it's always a good idea to check your work by hand to make sure you understand the steps involved in simplifying the expression.

Additional Resources

Khan Academy: Expanding and Simplifying Algebraic Expressions
Mathway: Expanding and Simplifying Algebraic Expressions
Wolfram Alpha: Expanding and Simplifying Algebraic Expressions

Final Answer

The final answer to the expression $(x+2)(x^2-2x+4)$ is $x^3 + 2x^2 + 8$ .