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Introduction

Polynomial expressions are a fundamental concept in algebra, and simplifying them is an essential skill for any math enthusiast. In this article, we will simplify the given polynomial expression by combining like terms and performing arithmetic operations. We will also provide a step-by-step guide on how to simplify polynomial expressions.

The Given Polynomial Expression

The given polynomial expression is:

$\left(-4x^2 + 17x + 9\right) - \left(-13x^2 + 5x + 16\right)$

Step 1: Distribute the Negative Sign

To simplify the given expression, we need to distribute the negative sign to each term inside the second set of parentheses.

$\left(-4x^2 + 17x + 9\right) - \left(-13x^2 + 5x + 16\right)$

$= -4x^2 + 17x + 9 + 13x^2 - 5x - 16$

Step 2: Combine Like Terms

Now, we need to combine like terms, which means combining terms with the same variable and exponent.

$= -4x^2 + 17x + 9 + 13x^2 - 5x - 16$

$= \left(-4x^2 + 13x^2\right) + \left(17x - 5x\right) + \left(9 - 16\right)$

$= 9x^2 + 12x - 7$

The Simplified Polynomial Expression

The simplified polynomial expression is:

$9x^2 + 12x - 7$

Conclusion

Simplifying polynomial expressions is an essential skill for any math enthusiast. By following the steps outlined in this article, we can simplify complex polynomial expressions and arrive at a simplified form. Remember to distribute the negative sign and combine like terms to simplify polynomial expressions.

Tips and Tricks

• When simplifying polynomial expressions, always start by distributing the negative sign to each term inside the second set of parentheses.
• Combine like terms by adding or subtracting terms with the same variable and exponent.
• Use the distributive property to simplify polynomial expressions.

Example Problems

• Simplify the polynomial expression: $\left(2x^2 + 5x + 3\right) - \left(x^2 + 2x + 1\right)$
• Simplify the polynomial expression: $\left(-3x^2 + 2x - 1\right) + \left(4x^2 - 3x + 2\right)$

Answer Key

• Simplified polynomial expression: $x^2 + 3x + 2$
• Simplified polynomial expression: $x^2 - x + 1$

Final Thoughts

Introduction

In our previous article, we simplified the given polynomial expression by combining like terms and performing arithmetic operations. In this article, we will answer some frequently asked questions (FAQs) related to simplifying polynomial expressions.

Q&A

Q: What is a polynomial expression?

A: A polynomial expression is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. It can be written in the form:

$a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$

where $a_n, a_{n-1}, \ldots, a_1, a_0$ are coefficients, and $x$ is the variable.

Q: What is the distributive property?

A: The distributive property is a mathematical property that allows us to distribute a coefficient to each term inside a set of parentheses. For example:

$2(x + y) = 2x + 2y$

Q: How do I simplify a polynomial expression?

A: To simplify a polynomial expression, follow these steps:

1. Distribute the negative sign to each term inside the second set of parentheses.
2. Combine like terms by adding or subtracting terms with the same variable and exponent.
3. Use the distributive property to simplify the expression.

Q: What is the difference between a polynomial expression and an algebraic expression?

A: A polynomial expression is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. An algebraic expression, on the other hand, can include any mathematical operation, such as division, exponentiation, and roots.

Q: Can I simplify a polynomial expression with negative coefficients?

A: Yes, you can simplify a polynomial expression with negative coefficients. When simplifying, remember to distribute the negative sign to each term inside the second set of parentheses.

Q: How do I know if a polynomial expression is simplified?

A: A polynomial expression is simplified when there are no like terms left to combine. In other words, when all the terms have different variables and exponents, the expression is simplified.

Q: Can I simplify a polynomial expression with variables in the denominator?

A: No, you cannot simplify a polynomial expression with variables in the denominator. To simplify such an expression, you need to rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator.

Q: What is the importance of simplifying polynomial expressions?

A: Simplifying polynomial expressions is essential in algebra and mathematics. It helps us to:

• Solve equations and inequalities
• Graph functions
• Find the roots of a polynomial
• Perform calculations and operations

Conclusion

Simplifying polynomial expressions is an essential skill for any math enthusiast. By following the steps outlined in this article and answering the FAQs, you can become proficient in simplifying polynomial expressions and tackle even the most complex math problems.

Tips and Tricks

• Always start by distributing the negative sign to each term inside the second set of parentheses.
• Combine like terms by adding or subtracting terms with the same variable and exponent.
• Use the distributive property to simplify polynomial expressions.
• Remember to rationalize the denominator when simplifying expressions with variables in the denominator.

Example Problems

• Simplify the polynomial expression: $\left(2x^2 + 5x + 3\right) - \left(x^2 + 2x + 1\right)$
• Simplify the polynomial expression: $\left(-3x^2 + 2x - 1\right) + \left(4x^2 - 3x + 2\right)$

Answer Key

• Simplified polynomial expression: $x^2 + 3x + 2$
• Simplified polynomial expression: $x^2 - x + 1$

Final Thoughts

Simplifying polynomial expressions is an essential skill for any math enthusiast. By following the steps outlined in this article and answering the FAQs, you can become proficient in simplifying polynomial expressions and tackle even the most complex math problems. With practice and patience, you can become a master of simplifying polynomial expressions and excel in algebra and mathematics.