Simplifying PolynomialsWhich Polynomial Correctly Combines The Like Terms And Puts The Given Polynomial In Standard Form?Given Polynomial: $\[ -5x^3y^3 + 8x^4y^2 - Xy^5 - 2x^2y^4 + 8x^6 + 3x^2y^4 - 6xy^5 \\]Options:A. \[$-7xy^5 + 5x^2y^4 -

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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, with a focus on combining like terms and putting the given polynomial in standard form.

What are Polynomials?

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 a way of writing the polynomial with the terms arranged in descending order of their exponents. For example, the polynomial x3+2x2−3x+1x^3 + 2x^2 - 3x + 1 is in standard form.

Combining Like Terms

Like terms are terms that have the same variable and exponent. Combining like terms involves adding or subtracting the coefficients of these terms. For example, in the polynomial 2x2+3x2−4x22x^2 + 3x^2 - 4x^2, the like terms are 2x22x^2, 3x23x^2, and −4x2-4x^2. Combining these terms gives us x2x^2.

Simplifying the Given Polynomial

Now, let's apply the concept of combining like terms to the given polynomial:

{ -5x^3y^3 + 8x^4y^2 - xy^5 - 2x^2y^4 + 8x^6 + 3x^2y^4 - 6xy^5 \}

To simplify this polynomial, we need to combine the like terms. We can start by grouping the terms with the same exponent.

Grouping Terms with the Same Exponent

Let's group the terms with the same exponent:

  • Terms with x6x^6: 8x68x^6
  • Terms with x4x^4: 8x4y28x^4y^2
  • Terms with x3x^3: −5x3y3-5x^3y^3
  • Terms with x2x^2: −2x2y4+3x2y4-2x^2y^4 + 3x^2y^4
  • Terms with xx: −xy5−6xy5-xy^5 - 6xy^5
  • Terms with no xx: None

Combining Like Terms

Now, let's combine the like terms:

  • Terms with x6x^6: 8x68x^6
  • Terms with x4x^4: 8x4y28x^4y^2
  • Terms with x3x^3: −5x3y3-5x^3y^3
  • Terms with x2x^2: x2y4x^2y^4
  • Terms with xx: −7xy5-7xy^5
  • Terms with no xx: None

Writing the Simplified Polynomial

Now that we have combined the like terms, we can write the simplified polynomial:

{ 8x^6 + 8x^4y^2 - 5x^3y^3 + x^2y^4 - 7xy^5 \}

Conclusion

Simplifying polynomials involves combining like terms and putting the given polynomial in standard form. By following the steps outlined in this article, you can simplify any polynomial and write it in standard form. Remember to group the terms with the same exponent and combine the like terms to get the simplified polynomial.

Frequently Asked Questions

Q: What is the standard form of a polynomial?

A: The standard form of a polynomial is a way of writing the polynomial with the terms arranged in descending order of their exponents.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the terms with the same variable and exponent.

Q: What is the difference between like terms and unlike terms?

A: Like terms are terms that have the same variable and exponent, while unlike terms are terms that have different variables or exponents.

Q: Can I simplify a polynomial with no like terms?

A: Yes, you can simplify a polynomial with no like terms by simply writing it in standard form.

Example Problems

Problem 1

Simplify the polynomial: 2x2+3x2−4x22x^2 + 3x^2 - 4x^2

Solution

Combine the like terms: x2x^2

Problem 2

Simplify the polynomial: x3+2x2−3x+1x^3 + 2x^2 - 3x + 1

Solution

The polynomial is already in standard form, so no simplification is needed.

Problem 3

Simplify the polynomial: −5x3y3+8x4y2−xy5−2x2y4+8x6+3x2y4−6xy5-5x^3y^3 + 8x^4y^2 - xy^5 - 2x^2y^4 + 8x^6 + 3x^2y^4 - 6xy^5

Solution

Combine the like terms: 8x6+8x4y2−5x3y3+x2y4−7xy58x^6 + 8x^4y^2 - 5x^3y^3 + x^2y^4 - 7xy^5

Practice Problems

Problem 1

Simplify the polynomial: x2+2x2−3x2x^2 + 2x^2 - 3x^2

Problem 2

Simplify the polynomial: x3+2x2−3x+1x^3 + 2x^2 - 3x + 1

Problem 3

Simplify the polynomial: −5x3y3+8x4y2−xy5−2x2y4+8x6+3x2y4−6xy5-5x^3y^3 + 8x^4y^2 - xy^5 - 2x^2y^4 + 8x^6 + 3x^2y^4 - 6xy^5

Answer Key

Problem 1

x2x^2

Problem 2

The polynomial is already in standard form, so no simplification is needed.

Problem 3

8x6+8x4y2−5x3y3+x2y4−7xy58x^6 + 8x^4y^2 - 5x^3y^3 + x^2y^4 - 7xy^5

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In our previous article, we explored the process of simplifying polynomials, with a focus on combining like terms and putting the given polynomial in standard form. In this article, we will answer some frequently asked questions about simplifying polynomials.

Q&A: Simplifying Polynomials

Q: What is the standard form of a polynomial?

A: The standard form of a polynomial is a way of writing the polynomial with the terms arranged in descending order of their exponents.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the terms with the same variable and exponent.

Q: What is the difference between like terms and unlike terms?

A: Like terms are terms that have the same variable and exponent, while unlike terms are terms that have different variables or exponents.

Q: Can I simplify a polynomial with no like terms?

A: Yes, you can simplify a polynomial with no like terms by simply writing it in standard form.

Q: How do I know if a term is a like term or an unlike term?

A: To determine if a term is a like term or an unlike term, you need to compare the variables and exponents of the term with the other terms in the polynomial.

Q: Can I simplify a polynomial with variables and constants?

A: Yes, you can simplify a polynomial with variables and constants by combining the like terms and writing the polynomial in standard form.

Q: How do I simplify a polynomial with negative coefficients?

A: To simplify a polynomial with negative coefficients, you need to combine the like terms and write the polynomial in standard form.

Q: Can I simplify a polynomial with fractional coefficients?

A: Yes, you can simplify a polynomial with fractional coefficients by combining the like terms and writing the polynomial in standard form.

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

A: To simplify a polynomial with multiple variables, you need to combine the like terms and write the polynomial in standard form.

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

A: Yes, you can simplify a polynomial with a variable in the denominator by combining the like terms and writing the polynomial in standard form.

Tips and Tricks

Tip 1: Start by grouping the terms with the same exponent.

This will make it easier to combine the like terms.

Tip 2: Use a table or chart to keep track of the like terms.

This will help you to avoid mistakes and ensure that you combine the correct terms.

Tip 3: Check your work by plugging in values for the variables.

This will help you to verify that your simplified polynomial is correct.

Tip 4: Use a calculator or computer program to check your work.

This will help you to verify that your simplified polynomial is correct and to identify any mistakes.

Common Mistakes

Mistake 1: Failing to combine like terms.

This can result in a polynomial that is not in standard form.

Mistake 2: Combining unlike terms.

This can result in a polynomial that is not in standard form.

Mistake 3: Failing to check the work.

This can result in a polynomial that is not correct.

Mistake 4: Using a calculator or computer program incorrectly.

This can result in a polynomial that is not correct.

Conclusion

Simplifying polynomials is an essential skill for any math enthusiast. By following the steps outlined in this article and using the tips and tricks provided, you can simplify any polynomial and write it in standard form. Remember to start by grouping the terms with the same exponent, use a table or chart to keep track of the like terms, and check your work by plugging in values for the variables.

Frequently Asked Questions

Q: What is the standard form of a polynomial?

A: The standard form of a polynomial is a way of writing the polynomial with the terms arranged in descending order of their exponents.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the terms with the same variable and exponent.

Q: What is the difference between like terms and unlike terms?

A: Like terms are terms that have the same variable and exponent, while unlike terms are terms that have different variables or exponents.

Q: Can I simplify a polynomial with no like terms?

A: Yes, you can simplify a polynomial with no like terms by simply writing it in standard form.

Example Problems

Problem 1

Simplify the polynomial: 2x2+3x2−4x22x^2 + 3x^2 - 4x^2

Solution

Combine the like terms: x2x^2

Problem 2

Simplify the polynomial: x3+2x2−3x+1x^3 + 2x^2 - 3x + 1

Solution

The polynomial is already in standard form, so no simplification is needed.

Problem 3

Simplify the polynomial: −5x3y3+8x4y2−xy5−2x2y4+8x6+3x2y4−6xy5-5x^3y^3 + 8x^4y^2 - xy^5 - 2x^2y^4 + 8x^6 + 3x^2y^4 - 6xy^5

Solution

Combine the like terms: 8x6+8x4y2−5x3y3+x2y4−7xy58x^6 + 8x^4y^2 - 5x^3y^3 + x^2y^4 - 7xy^5

Practice Problems

Problem 1

Simplify the polynomial: x2+2x2−3x2x^2 + 2x^2 - 3x^2

Problem 2

Simplify the polynomial: x3+2x2−3x+1x^3 + 2x^2 - 3x + 1

Problem 3

Simplify the polynomial: −5x3y3+8x4y2−xy5−2x2y4+8x6+3x2y4−6xy5-5x^3y^3 + 8x^4y^2 - xy^5 - 2x^2y^4 + 8x^6 + 3x^2y^4 - 6xy^5

Answer Key

Problem 1

x2x^2

Problem 2

The polynomial is already in standard form, so no simplification is needed.

Problem 3

8x6+8x4y2−5x3y3+x2y4−7xy58x^6 + 8x^4y^2 - 5x^3y^3 + x^2y^4 - 7xy^5