Factor Out The Greatest Common Factor From The Following Polynomial:$\[ 15a^8b^8 - 25a^9b^9 + 25ab - 10a^9b \\]A. \[$ 15a^0b^0 - 25a^3b^y + 25ab - 10a^y B = \$\](Type Your Answer In Factored Form.)

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Introduction

Factoring out the greatest common factor (GCF) from a polynomial is an essential skill in algebra. It involves identifying the common factors among the terms of a polynomial and expressing the polynomial as a product of the GCF and a new polynomial. In this article, we will factor out the greatest common factor from the given polynomial: ${ 15a8b8 - 25a9b9 + 25ab - 10a^9b }$

Understanding the Greatest Common Factor

The greatest common factor (GCF) of a set of numbers is the largest number that divides each of the numbers in the set without leaving a remainder. In the context of polynomials, the GCF is the term that is common to all the terms in the polynomial. To factor out the GCF, we need to identify the common factors among the terms and express the polynomial as a product of the GCF and a new polynomial.

Step 1: Identify the Common Factors

To factor out the GCF, we need to identify the common factors among the terms of the polynomial. The given polynomial is: ${ 15a8b8 - 25a9b9 + 25ab - 10a^9b }$

The common factors among the terms are:

  • −5-5
  • a8b8a^8b^8
  • a9ba^9b

Step 2: Factor Out the GCF

Now that we have identified the common factors, we can factor out the GCF. The GCF of the polynomial is −5a8b8-5a^8b^8. We can factor out this term from each of the terms in the polynomial:

−5a8b8(3)−5a8b8(5a1b1)+5a8b8(5a−8b−8)−5a8b8(2a1b1)-5a^8b^8(3) - 5a^8b^8(5a^1b^1) + 5a^8b^8(5a^{-8}b^{-8}) - 5a^8b^8(2a^1b^1)

Step 3: Simplify the Expression

Now that we have factored out the GCF, we can simplify the expression by combining like terms:

−15a8b8−25a9b9+25ab−10a9b-15a^8b^8 - 25a^9b^9 + 25ab - 10a^9b

Step 4: Write the Final Answer

The final answer is: −5a8b8(3−5a1b1+5a−8b−8−2a1b1)\boxed{-5a^8b^8(3 - 5a^1b^1 + 5a^{-8}b^{-8} - 2a^1b^1)}

However, we can simplify this expression further by combining like terms:

−5a8b8(3−5a1b1−2a1b1+5a−8b−8)\boxed{-5a^8b^8(3 - 5a^1b^1 - 2a^1b^1 + 5a^{-8}b^{-8})}

Conclusion

Factoring out the greatest common factor from a polynomial is an essential skill in algebra. It involves identifying the common factors among the terms of a polynomial and expressing the polynomial as a product of the GCF and a new polynomial. In this article, we factored out the greatest common factor from the given polynomial and simplified the expression to obtain the final answer.

Common Mistakes to Avoid

When factoring out the greatest common factor from a polynomial, there are several common mistakes to avoid:

  • Not identifying the common factors: Make sure to identify the common factors among the terms of the polynomial.
  • Not factoring out the GCF: Make sure to factor out the GCF from each of the terms in the polynomial.
  • Not simplifying the expression: Make sure to simplify the expression by combining like terms.

Practice Problems

Here are some practice problems to help you practice factoring out the greatest common factor from a polynomial:

  • Factor out the GCF from the polynomial: 2x3y3−3x2y2+4xy−52x^3y^3 - 3x^2y^2 + 4xy - 5
  • Factor out the GCF from the polynomial: 3a4b4−2a3b3+5ab−43a^4b^4 - 2a^3b^3 + 5ab - 4
  • Factor out the GCF from the polynomial: 4x5y5−3x4y4+2xy−14x^5y^5 - 3x^4y^4 + 2xy - 1

Conclusion

Q: What is the greatest common factor (GCF) of a polynomial?

A: The greatest common factor (GCF) of a polynomial is the largest term that divides each of the terms in the polynomial without leaving a remainder.

Q: How do I identify the common factors among the terms of a polynomial?

A: To identify the common factors among the terms of a polynomial, you need to look for the terms that are common to all the terms in the polynomial. These terms can be constants, variables, or a combination of both.

Q: What is the first step in factoring out the GCF from a polynomial?

A: The first step in factoring out the GCF from a polynomial is to identify the common factors among the terms of the polynomial.

Q: How do I factor out the GCF from a polynomial?

A: To factor out the GCF from a polynomial, you need to multiply each term in the polynomial by the GCF. This will result in a new polynomial that has the GCF factored out.

Q: What is the next step after factoring out the GCF from a polynomial?

A: After factoring out the GCF from a polynomial, you need to simplify the expression by combining like terms.

Q: What are some common mistakes to avoid when factoring out the GCF from a polynomial?

A: Some common mistakes to avoid when factoring out the GCF from a polynomial include:

  • Not identifying the common factors among the terms of the polynomial.
  • Not factoring out the GCF from each of the terms in the polynomial.
  • Not simplifying the expression by combining like terms.

Q: How do I simplify the expression after factoring out the GCF from a polynomial?

A: To simplify the expression after factoring out the GCF from a polynomial, you need to combine like terms. This involves adding or subtracting the coefficients of the terms that have the same variable and exponent.

Q: What is the final answer after factoring out the GCF from a polynomial?

A: The final answer after factoring out the GCF from a polynomial is the simplified expression that has the GCF factored out.

Q: Can you provide some examples of polynomials that can be factored out the GCF?

A: Yes, here are some examples of polynomials that can be factored out the GCF:

  • 2x3y3−3x2y2+4xy−52x^3y^3 - 3x^2y^2 + 4xy - 5
  • 3a4b4−2a3b3+5ab−43a^4b^4 - 2a^3b^3 + 5ab - 4
  • 4x5y5−3x4y4+2xy−14x^5y^5 - 3x^4y^4 + 2xy - 1

Q: How do I practice factoring out the GCF from a polynomial?

A: To practice factoring out the GCF from a polynomial, you can try the following:

  • Start with simple polynomials and gradually move on to more complex ones.
  • Use online resources or worksheets to practice factoring out the GCF from polynomials.
  • Try to factor out the GCF from polynomials on your own and then check your answers with a calculator or a teacher.

Conclusion

Factoring out the greatest common factor from a polynomial is an essential skill in algebra. It involves identifying the common factors among the terms of a polynomial and expressing the polynomial as a product of the GCF and a new polynomial. In this article, we provided answers to some common questions about factoring out the GCF from a polynomial and provided some examples of polynomials that can be factored out the GCF.