Simplify The Polynomial Below. Write Your Answer In Standard Form.$\[ (5x^2 - 6x + 2) + (x^2 + 9x - 4) = \square \\]

Introduction

Polynomials are a fundamental concept in algebra, and simplifying them is an essential skill for any math enthusiast. In this article, we will explore the process of simplifying polynomials, using the given polynomial as an example. We will break down the steps involved in simplifying the polynomial and provide a clear explanation of each step.

What is a Polynomial?

A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. Polynomials can be written in various forms, including standard form, factored form, and expanded form.

Standard Form of a Polynomial

The standard form of a polynomial is written with the terms in descending order of their exponents. For example, the polynomial $3x^2 + 2x - 4$ is in standard form.

Simplifying the Given Polynomial

The given polynomial is $(5x^2 - 6x + 2) + (x^2 + 9x - 4)$. To simplify this polynomial, we need to combine like terms.

Like Terms

Like terms are terms that have the same variable and exponent. In the given polynomial, the like terms are:

• $5x^2$ and $x^2$
• $-6x$ and $9x$
• $2$ and $-4$

Combining Like Terms

To combine like terms, we add or subtract their coefficients. In this case, we add the coefficients of the like terms.

• $5x^2 + x^2 = 6x^2$
• $-6x + 9x = 3x$
• $2 - 4 = -2$

Simplifying the Polynomial

Now that we have combined the like terms, we can simplify the polynomial by adding the simplified terms.

$(5x^2 - 6x + 2) + (x^2 + 9x - 4) = 6x^2 + 3x - 2$

Conclusion

Simplifying polynomials is an essential skill in algebra, and it requires a clear understanding of like terms and how to combine them. By following the steps outlined in this article, you can simplify any polynomial and express it in standard form.

Example Problems

• Simplify the polynomial $(2x^2 + 3x - 1) + (x^2 - 2x + 4)$.
• Simplify the polynomial $(4x^2 - 2x + 3) + (2x^2 + 5x - 1)$.

Tips and Tricks

• When simplifying polynomials, always start by combining like terms.
• Use the distributive property to expand expressions and combine like terms.
• Check your work by plugging in values for the variable and simplifying the expression.

Common Mistakes

• Failing to combine like terms.
• Adding or subtracting coefficients incorrectly.
• Not checking work by plugging in values for the variable.

Real-World Applications

Simplifying polynomials has many real-world applications, including:

• Science and Engineering: Polynomials are used to model real-world phenomena, such as the motion of objects and the behavior of electrical circuits.
• Computer Science: Polynomials are used in computer algorithms, such as sorting and searching.
• Economics: Polynomials are used to model economic systems and make predictions about future trends.

Conclusion

Introduction

In our previous article, we explored the process of simplifying polynomials, using the given polynomial as an example. In this article, we will answer some frequently asked questions about simplifying polynomials, providing a clear explanation of each concept.

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

A: A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. An expression, on the other hand, can be any combination of variables, coefficients, and operations.

Q: How do I know if two terms are like terms?

A: Two terms are like terms if they have the same variable and exponent. For example, $2x^2$ and $5x^2$ are like terms because they both have the variable $x$ and the exponent $2$.

Q: Can I combine terms with different exponents?

A: No, you cannot combine terms with different exponents. For example, $2x^2$ and $3x$ cannot be combined because they have different exponents.

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

A: To simplify a polynomial with multiple variables, you need to combine like terms for each variable. For example, if you have the polynomial $2x^2y + 3x^2y - 4xy$, you can combine the like terms for the variable $x$ and the variable $y$.

Q: Can I simplify a polynomial with negative coefficients?

A: Yes, you can simplify a polynomial with negative coefficients. When combining like terms, you need to add or subtract the coefficients, regardless of whether they are positive or negative.

Q: How do I check my work when simplifying a polynomial?

A: To check your work, you can plug in values for the variable and simplify the expression. For example, if you have the polynomial $2x^2 + 3x - 4$ and you plug in $x = 2$, you should get the simplified expression $2(2)^2 + 3(2) - 4 = 12 + 6 - 4 = 14$.

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

A: No, you cannot simplify a polynomial with a variable in the denominator. For example, the expression $\frac{2x}{x}$ cannot be simplified because it has a variable in the denominator.

Q: How do I simplify a polynomial with a fraction coefficient?

A: To simplify a polynomial with a fraction coefficient, you need to multiply the numerator and denominator by the same value to eliminate the fraction. For example, if you have the polynomial $2\frac{1}{2}x^2 + 3x - 4$, you can multiply the numerator and denominator by $2$ to get $5x^2 + 3x - 4$.

Q: Can I simplify a polynomial with a negative exponent?

A: No, you cannot simplify a polynomial with a negative exponent. For example, the expression $2x^{-2}$ cannot be simplified because it has a negative exponent.

Conclusion

Simplifying polynomials is an essential skill in algebra, and it requires a clear understanding of like terms and how to combine them. By following the steps outlined in this article, you can simplify any polynomial and express it in standard form. Remember to always check your work by plugging in values for the variable and to use the distributive property to expand expressions and combine like terms.

Example Problems

• Simplify the polynomial $(2x^2 + 3x - 4) + (x^2 - 2x + 5)$.
• Simplify the polynomial $(4x^2 - 2x + 3) + (2x^2 + 5x - 1)$.

Tips and Tricks

• When simplifying polynomials, always start by combining like terms.
• Use the distributive property to expand expressions and combine like terms.
• Check your work by plugging in values for the variable.

Common Mistakes

• Failing to combine like terms.
• Adding or subtracting coefficients incorrectly.
• Not checking work by plugging in values for the variable.

Real-World Applications

Simplifying polynomials has many real-world applications, including:

• Science and Engineering: Polynomials are used to model real-world phenomena, such as the motion of objects and the behavior of electrical circuits.
• Computer Science: Polynomials are used in computer algorithms, such as sorting and searching.
• Economics: Polynomials are used to model economic systems and make predictions about future trends.