Write The Following Expression As A Simplified Polynomial In Standard Form.$ 2(x-2)^2 - 3(x-2) - 4 }$Answer { \square$ $Submit Answer

Introduction

Polynomials are a fundamental concept in algebra, and simplifying them is a crucial skill for any math enthusiast. In this article, we will explore how to simplify a given polynomial expression using the distributive property and combining like terms. We will use the expression $2(x-2)^2 - 3(x-2) - 4$ as an example and break it down into a simplified polynomial in standard form.

Understanding the Expression

Before we start simplifying the expression, let's take a closer look at what we're dealing with. The given expression is:

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

This expression consists of three terms:

1. $2(x-2)^2$
2. $-3(x-2)$
3. $-4$

Step 1: Expand the First Term

The first term is $2(x-2)^2$. To simplify this term, we need to expand the squared expression using the formula $(a-b)^2 = a^2 - 2ab + b^2$. In this case, $a = x$ and $b = 2$.

$$(x-2)^2 = x^2 - 2x(2) + 2^2$$

$$= x^2 - 4x + 4$$

Now, we can multiply the coefficient $2$ with the expanded expression:

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

$$= 2x^2 - 8x + 8$$

Step 2: Distribute the Negative Sign

The second term is $-3(x-2)$. To simplify this term, we need to distribute the negative sign to the expression inside the parentheses:

$$-3(x-2) = -3x + 6$$

Step 3: Combine Like Terms

Now that we have expanded and distributed the terms, we can combine like terms to simplify the expression. We have three terms:

1. $2x^2 - 8x + 8$
2. $-3x + 6$
3. $-4$

To combine like terms, we need to group the terms with the same variable and exponent. In this case, we have two terms with the variable $x$:

1. $-8x$ (from the first term)
2. $-3x$ (from the second term)

We can combine these two terms by adding their coefficients:

$$(-8x) + (-3x) = -11x$$

Now, we can rewrite the expression with the combined like terms:

$$2x^2 - 11x + 8 + 6 - 4$$

$$= 2x^2 - 11x + 10$$

Conclusion

In this article, we simplified the given polynomial expression $2(x-2)^2 - 3(x-2) - 4$ using the distributive property and combining like terms. We expanded the first term, distributed the negative sign, and combined like terms to arrive at the simplified polynomial in standard form: $2x^2 - 11x + 10$. This process demonstrates the importance of understanding and applying algebraic properties to simplify complex expressions.

Final Answer

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Introduction

In our previous article, we explored how to simplify a given polynomial expression using the distributive property and combining like terms. In this article, we will answer some frequently asked questions about simplifying polynomials, providing additional insights and examples to help you master this essential algebraic skill.

Q&A

Q: What is the distributive property, and how is it used in simplifying polynomials?

A: The distributive property is a fundamental concept in algebra that allows us to multiply a single term to multiple terms inside parentheses. In the context of simplifying polynomials, the distributive property is used to expand expressions like $(x-2)^2$ or $-3(x-2)$.

For example, consider the expression $2(x-2)^2$. Using the distributive property, we can expand the squared expression as follows:

$$(x-2)^2 = x^2 - 2x(2) + 2^2$$

$$= x^2 - 4x + 4$$

Now, we can multiply the coefficient $2$ with the expanded expression:

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

$$= 2x^2 - 8x + 8$$

Q: How do I combine like terms in a polynomial expression?

A: Combining like terms is a crucial step in simplifying polynomial expressions. To combine like terms, you need to group the terms with the same variable and exponent. In the expression $2x^2 - 8x + 8 - 3x + 6 - 4$, we have two terms with the variable $x$:

1. $-8x$ (from the first term)
2. $-3x$ (from the second term)

We can combine these two terms by adding their coefficients:

$$(-8x) + (-3x) = -11x$$

Now, we can rewrite the expression with the combined like terms:

$$2x^2 - 11x + 8 + 6 - 4$$

$$= 2x^2 - 11x + 10$$

Q: What is the standard form of a polynomial expression?

A: The standard form of a polynomial expression is a way of writing the expression with the terms arranged in descending order of their exponents. For example, the expression $2x^2 - 11x + 10$ is in standard form because the terms are arranged in descending order of their exponents:

1. $2x^2$ (exponent: 2)
2. $-11x$ (exponent: 1)
3. $10$ (exponent: 0)

Q: How do I simplify a polynomial expression with multiple variables?

A: Simplifying a polynomial expression with multiple variables requires careful attention to the distributive property and combining like terms. For example, consider the expression $2(x-2)^2 - 3(x-2) - 4$. To simplify this expression, we need to expand the squared expression and distribute the negative sign:

$$(x-2)^2 = x^2 - 2x(2) + 2^2$$

$$= x^2 - 4x + 4$$

Now, we can multiply the coefficient $2$ with the expanded expression:

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

$$= 2x^2 - 8x + 8$$

Next, we need to distribute the negative sign to the expression inside the parentheses:

$$-3(x-2) = -3x + 6$$

Finally, we can combine like terms to simplify the expression:

$$2x^2 - 8x + 8 - 3x + 6 - 4$$

$$= 2x^2 - 11x + 10$$

Q: What are some common mistakes to avoid when simplifying polynomial expressions?

A: When simplifying polynomial expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

1. Not distributing the negative sign: Make sure to distribute the negative sign to the expression inside the parentheses.
2. Not combining like terms: Make sure to combine like terms to simplify the expression.
3. Not arranging terms in standard form: Make sure to arrange the terms in descending order of their exponents.

By avoiding these common mistakes, you can ensure that your polynomial expressions are simplified correctly.

Conclusion

In this article, we answered some frequently asked questions about simplifying polynomials, providing additional insights and examples to help you master this essential algebraic skill. By understanding the distributive property, combining like terms, and arranging terms in standard form, you can simplify polynomial expressions with confidence. Remember to avoid common mistakes and practice, practice, practice to become a pro at simplifying polynomials!