Expanding the Expression Using the Distributive Property

Oct 13, 2025 by talynt.org 120 views

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Introduction

In this article, we will focus on expanding and simplifying the given polynomial expression ${$ (x-2)(2x+3)(x+5)$}$ to the form ${$ ax^3+bx2+cx+d$}$, where ${$ a$}$ and ${$ b$}$ are positive integers, and ${$ c$}$ and ${$ d$}$ are negative integers. We will use the distributive property and the FOIL method to expand the given expression.

Expanding the Expression Using the Distributive Property

The distributive property states that for any real numbers ${$ a$, [ $b\$, and \[$ c$, the following equation holds:

[ $a(b+c) = ab + ac\$}$

We can use this property to expand the given expression by multiplying each term in the first parentheses by each term in the second parentheses, and then multiplying each term in the result by each term in the third parentheses.

Step 1: Multiply the First Two Parentheses

Using the distributive property, we can multiply the first two parentheses as follows:

${$ (x-2)(2x+3) = x(2x+3) - 2(2x+3)$}$

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

${$ = 2x^2 - x - 6$}$

Step 2: Multiply the Result by the Third Parentheses

Now, we can multiply the result by the third parentheses as follows:

${$ (2x^2 - x - 6)(x+5) = 2x^2(x+5) - x(x+5) - 6(x+5)$}$

${$ = 2x^3 + 10x^2 - x^2 - 5x - 6x - 30$}$

${$ = 2x^3 + 9x^2 - 11x - 30$}$

Conclusion

We have successfully expanded and simplified the given polynomial expression to the form ${$ ax^3+bx2+cx+d$}$, where ${$ a = 2$, [ $b = 9\$, \[$ c = -11$, and [ $d = -30\$. Therefore, the values of \[$ a$, [ $b\$, \[$ c$, and [ $d\$}$ are ${$2$, [ $9\$, \[$ -11$, and [$-30$, respectively.

Final Answer

The final answer is [$2x^3 + 9x^2 - 11x - 30$.

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Introduction

In our previous article, we expanded and simplified the given polynomial expression [ $(x-2)(2x+3)(x+5)\$}$ to the form ${$ ax^3+bx2+cx+d$}$, where ${$ a$}$ and ${$ b$}$ are positive integers, and ${$ c$}$ and ${$ d$}$ are negative integers. In this article, we will answer some frequently asked questions related to the expansion and simplification of polynomial expressions.

Q&A

Q: What is the distributive property, and how is it used in expanding polynomial expressions?

A: The distributive property is a mathematical concept that states that for any real numbers ${$ a$, [ $b\$, and \[$ c$, the following equation holds:

[ $a(b+c) = ab + ac\$}$

This property is used in expanding polynomial expressions by multiplying each term in one parentheses by each term in another parentheses.

Q: How do I expand a polynomial expression using the FOIL method?

A: The FOIL method is a technique used to expand polynomial expressions by multiplying each term in the first parentheses by each term in the second parentheses, and then multiplying each term in the result by each term in the third parentheses. The FOIL method is an acronym that stands for "First, Outer, Inner, Last," which refers to the order in which the terms are multiplied.

Q: What is the difference between expanding and simplifying a polynomial expression?

A: Expanding a polynomial expression involves multiplying the terms together to get a single expression, while simplifying a polynomial expression involves combining like terms to get a simpler expression.

Q: How do I simplify a polynomial expression?

A: To simplify a polynomial expression, you need to combine like terms. Like terms are terms that have the same variable and exponent. For example, ${ $2x^2\$}$ and ${ $3x^2\$}$ are like terms, and can be combined as ${ $5x^2\$}$ .

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

A: Some common mistakes to avoid when expanding and simplifying polynomial expressions include:

Forgetting to distribute the terms correctly
Not combining like terms
Making errors when multiplying the terms together
Not checking the final expression for errors

Conclusion

Expanding and simplifying polynomial expressions can be a challenging task, but with practice and patience, you can become proficient in it. Remember to use the distributive property and the FOIL method to expand polynomial expressions, and combine like terms to simplify them. By following these steps and avoiding common mistakes, you can ensure that your final answer is accurate and correct.

Final Answer

The final answer is [$2x^3 + 9x^2 - 11x - 30$.