The Polynomial $8x^2 - 8x + 2 - 5 + X$ Is Simplified To $8x^2 - Gx - H$. What Are The Values Of \$g$[/tex\] And $h$?A. $g = -9$ And \$h = 7$[/tex\] B. $g = 9$ And

by ADMIN 176 views

Introduction

In algebra, polynomials are mathematical expressions consisting of variables and coefficients combined using addition, subtraction, and multiplication. Simplifying polynomials is an essential skill in mathematics, and it has numerous applications in various fields, including physics, engineering, and computer science. In this article, we will focus on simplifying a given polynomial and determining the values of its coefficients.

The Given Polynomial

The given polynomial is $8x^2 - 8x + 2 - 5 + x$. Our goal is to simplify this polynomial and express it in the form $8x^2 - gx - h$.

Step 1: Combine Like Terms

To simplify the polynomial, we need to combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have two like terms: $-8x$ and $x$. We can combine these terms by adding their coefficients:

−8x+x=−7x-8x + x = -7x

So, the polynomial becomes:

8x2−7x+2−58x^2 - 7x + 2 - 5

Step 2: Combine Constants

Next, we need to combine the constants. In this case, we have two constants: $2$ and $-5$. We can combine these constants by adding them:

2−5=−32 - 5 = -3

So, the polynomial becomes:

8x2−7x−38x^2 - 7x - 3

Step 3: Compare with the Simplified Form

Now, we need to compare the simplified polynomial with the given form $8x^2 - gx - h$. We can see that the coefficient of $x^2$ is the same in both forms, which is $8$. We also need to find the values of $g$ and $h$.

Determining the Value of g

To determine the value of $g$, we need to look at the coefficient of the $x$ term in the simplified polynomial. In this case, the coefficient of the $x$ term is $-7$. Therefore, the value of $g$ is:

g=−7g = -7

Determining the Value of h

To determine the value of $h$, we need to look at the constant term in the simplified polynomial. In this case, the constant term is $-3$. Therefore, the value of $h$ is:

h=3h = 3

Conclusion

In conclusion, the polynomial $8x^2 - 8x + 2 - 5 + x$ is simplified to $8x^2 - 7x - 3$. The values of $g$ and $h$ are $-7$ and $3$, respectively.

Answer

The correct answer is:

g = -7$ and $h = 3

Discussion

This problem is a great example of how to simplify polynomials and determine the values of their coefficients. It requires a good understanding of algebraic expressions and the ability to combine like terms. The problem also requires attention to detail, as small mistakes can lead to incorrect answers.

Related Topics

  • Algebraic expressions
  • Combining like terms
  • Simplifying polynomials
  • Determining coefficients

Further Reading

For more information on algebraic expressions and polynomial simplification, please refer to the following resources:

  • Khan Academy: Algebraic Expressions
  • Mathway: Simplifying Polynomials
  • Wolfram MathWorld: Polynomial Simplification
    The Polynomial Simplification Problem: Q&A =============================================

Introduction

In our previous article, we simplified the polynomial $8x^2 - 8x + 2 - 5 + x$ and determined the values of its coefficients. In this article, we will answer some frequently asked questions related to the problem.

Q: What is the coefficient of the x^2 term in the simplified polynomial?

A: The coefficient of the $x^2$ term in the simplified polynomial is $8$.

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

A: To combine like terms in a polynomial, you need to add or subtract the coefficients of the terms with the same variable raised to the same power.

Q: What is the value of g in the simplified polynomial?

A: The value of $g$ in the simplified polynomial is $-7$.

Q: What is the value of h in the simplified polynomial?

A: The value of $h$ in the simplified polynomial is $3$.

Q: How do I determine the values of g and h in a polynomial?

A: To determine the values of $g$ and $h$ in a polynomial, you need to look at the coefficients of the $x$ and constant terms, respectively.

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

A: A polynomial is a mathematical expression consisting of variables and coefficients combined using addition, subtraction, and multiplication. An algebraic expression is a general term that refers to any mathematical expression involving variables and constants.

Q: Can I simplify a polynomial with more than two terms?

A: Yes, you can simplify a polynomial with more than two terms by combining like terms.

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

A: A polynomial is simplified when there are no like terms that can be combined.

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

A: Some common mistakes to avoid when simplifying polynomials include:

  • Not combining like terms
  • Adding or subtracting coefficients incorrectly
  • Not checking for like terms

Conclusion

In conclusion, simplifying polynomials is an essential skill in mathematics, and it requires attention to detail and a good understanding of algebraic expressions. By following the steps outlined in this article, you can simplify polynomials and determine the values of their coefficients.

Related Topics

  • Algebraic expressions
  • Combining like terms
  • Simplifying polynomials
  • Determining coefficients

Further Reading

For more information on algebraic expressions and polynomial simplification, please refer to the following resources:

  • Khan Academy: Algebraic Expressions
  • Mathway: Simplifying Polynomials
  • Wolfram MathWorld: Polynomial Simplification

Practice Problems

Try simplifying the following polynomials:

  • 2x2+3x−4+2x2x^2 + 3x - 4 + 2x

  • x2−2x+3−4xx^2 - 2x + 3 - 4x

  • 3x2+2x−5+3x3x^2 + 2x - 5 + 3x

Answer Key

  • 2x2+5x−42x^2 + 5x - 4

  • x2−6x+3x^2 - 6x + 3

  • 3x2+5x−53x^2 + 5x - 5