Simplify The Expression:$ (x+2)(x^2 - 2x + 4) $

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Introduction

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In algebra, simplifying expressions is a crucial skill that helps us solve equations and inequalities. One of the most common methods of simplifying expressions is by using the distributive property, which states that for any real numbers a, b, and c, a(b + c) = ab + ac. In this article, we will use the distributive property to simplify the given expression: $(x+2)(x^2 - 2x + 4)$.

Understanding the Expression

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The given expression is a product of two binomials: $(x+2)$ and $(x^2 - 2x + 4)$. To simplify this expression, we need to multiply each term in the first binomial by each term in the second binomial. This process is called the FOIL method, which stands for First, Outer, Inner, Last.

FOIL Method

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The FOIL method is a step-by-step process for multiplying two binomials. Here's how it works:

1. Multiply the First terms: $x \cdot x^2$
2. Multiply the Outer terms: $x \cdot (-2x)$
3. Multiply the Inner terms: $2 \cdot x^2$
4. Multiply the Last terms: $2 \cdot (-2x)$
5. Combine like terms: $x^3 - 2x^2 + 2x^2 - 4x$

Simplifying the Expression

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Now that we have multiplied each term in the first binomial by each term in the second binomial, we can simplify the expression by combining like terms. In this case, we have two terms with the same variable and exponent: $-2x^2$ and $2x^2$. When we combine these terms, they cancel each other out, leaving us with:

$x^3 - 4x$

Final Answer

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The simplified expression is $x^3 - 4x$. This is the final answer to the problem.

Conclusion

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Simplifying expressions is an essential skill in algebra that helps us solve equations and inequalities. By using the distributive property and the FOIL method, we can simplify complex expressions and arrive at a final answer. In this article, we used the distributive property to simplify the expression $(x+2)(x^2 - 2x + 4)$ and arrived at the final answer $x^3 - 4x$.

Example Problems

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Here are a few example problems that demonstrate how to simplify expressions using the distributive property and the FOIL method:

Example 1

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Simplify the expression: $(x-3)(x^2 + 5x - 2)$

Solution

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Using the FOIL method, we multiply each term in the first binomial by each term in the second binomial:

$x \cdot x^2 = x^3$ $x \cdot 5x = 5x^2$ $x \cdot (-2) = -2x$ $-3 \cdot x^2 = -3x^2$ $-3 \cdot 5x = -15x$ $-3 \cdot (-2) = 6$

Combining like terms, we get:

$x^3 + 5x^2 - 3x^2 - 15x + 6$

Simplifying further, we get:

$x^3 + 2x^2 - 15x + 6$

Example 2

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Simplify the expression: $(x+4)(x^2 - 3x + 2)$

Solution

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Using the FOIL method, we multiply each term in the first binomial by each term in the second binomial:

$x \cdot x^2 = x^3$ $x \cdot (-3x) = -3x^2$ $x \cdot 2 = 2x$ $4 \cdot x^2 = 4x^2$ $4 \cdot (-3x) = -12x$ $4 \cdot 2 = 8$

Combining like terms, we get:

$x^3 - 3x^2 + 2x + 4x^2 - 12x + 8$

Simplifying further, we get:

$x^3 + x^2 - 10x + 8$

Tips and Tricks

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Here are a few tips and tricks to help you simplify expressions using the distributive property and the FOIL method:

• Make sure to multiply each term in the first binomial by each term in the second binomial.
• Use the FOIL method to simplify the expression.
• Combine like terms to simplify the expression further.
• Check your work by plugging in a value for x and simplifying the expression.

By following these tips and tricks, you can simplify complex expressions and arrive at a final answer.

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Introduction

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In our previous article, we simplified the expression $(x+2)(x^2 - 2x + 4)$ using the distributive property and the FOIL method. In this article, we will answer some common questions that students often have when simplifying expressions.

Q&A

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Q: What is the distributive property?

A: The distributive property is a mathematical concept that states that for any real numbers a, b, and c, a(b + c) = ab + ac. This means that we can multiply a single term by two or more terms inside a set of parentheses.

Q: How do I use the FOIL method to simplify expressions?

A: The FOIL method is a step-by-step process for multiplying two binomials. Here's how it works:

1. Multiply the First terms: $x \cdot x^2$
2. Multiply the Outer terms: $x \cdot (-2x)$
3. Multiply the Inner terms: $2 \cdot x^2$
4. Multiply the Last terms: $2 \cdot (-2x)$
5. Combine like terms: $x^3 - 2x^2 + 2x^2 - 4x$

Q: What are like terms?

A: Like terms are terms that have the same variable and exponent. For example, $2x^2$ and $-3x^2$ are like terms because they both have the variable $x$ and the exponent $2$.

Q: How do I combine like terms?

A: To combine like terms, we add or subtract the coefficients of the like terms. For example, if we have the expression $2x^2 + 3x^2$, we can combine the like terms by adding the coefficients: $2 + 3 = 5$. The resulting expression is $5x^2$.

Q: What if I have a term with a negative exponent?

A: If we have a term with a negative exponent, we can rewrite it as a fraction. For example, if we have the term $x^{-2}$, we can rewrite it as $\frac{1}{x^2}$.

Q: Can I simplify expressions with variables in the denominator?

A: Yes, we can simplify expressions with variables in the denominator. However, we need to be careful when multiplying or dividing expressions with variables in the denominator.

Q: How do I simplify expressions with fractions?

A: To simplify expressions with fractions, we can multiply the numerator and denominator by the same value to eliminate the fraction. For example, if we have the expression $\frac{x^2}{2}$, we can multiply the numerator and denominator by $2$ to get $x^2$.

Example Problems

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Here are a few example problems that demonstrate how to simplify expressions using the distributive property and the FOIL method:

Example 1

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Simplify the expression: $(x-3)(x^2 + 5x - 2)$

Solution

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Using the FOIL method, we multiply each term in the first binomial by each term in the second binomial:

$x \cdot x^2 = x^3$ $x \cdot 5x = 5x^2$ $x \cdot (-2) = -2x$ $-3 \cdot x^2 = -3x^2$ $-3 \cdot 5x = -15x$ $-3 \cdot (-2) = 6$

Combining like terms, we get:

$x^3 + 5x^2 - 3x^2 - 15x + 6$

Simplifying further, we get:

$x^3 + 2x^2 - 15x + 6$

Example 2

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Simplify the expression: $(x+4)(x^2 - 3x + 2)$

Solution

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Using the FOIL method, we multiply each term in the first binomial by each term in the second binomial:

$x \cdot x^2 = x^3$ $x \cdot (-3x) = -3x^2$ $x \cdot 2 = 2x$ $4 \cdot x^2 = 4x^2$ $4 \cdot (-3x) = -12x$ $4 \cdot 2 = 8$

Combining like terms, we get:

$x^3 - 3x^2 + 2x + 4x^2 - 12x + 8$

Simplifying further, we get:

$x^3 + x^2 - 10x + 8$

Tips and Tricks

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Here are a few tips and tricks to help you simplify expressions using the distributive property and the FOIL method:

• Make sure to multiply each term in the first binomial by each term in the second binomial.
• Use the FOIL method to simplify the expression.
• Combine like terms to simplify the expression further.
• Check your work by plugging in a value for x and simplifying the expression.

By following these tips and tricks, you can simplify complex expressions and arrive at a final answer.