Factor The Greatest Common Factor (GCF) From The Polynomial: { -9y^2 + 12y$}$A. { -3y(3y - 4)$}$B. ${ 3(-3y^2 - 4y)\$} C. ${ 9y(-3y^2 + 4)\$} D. { Y(-9y + 12)$}$

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Understanding the Greatest Common Factor (GCF)

The greatest common factor (GCF) of a set of numbers is the largest number that divides each of the numbers without leaving a remainder. In the context of polynomials, the GCF is the largest polynomial that divides each term of the polynomial without leaving a remainder. Factoring the GCF from a polynomial is an essential step in simplifying and solving polynomial equations.

Identifying the GCF of the Polynomial

To factor the GCF from the given polynomial, we need to identify the largest polynomial that divides each term of the polynomial without leaving a remainder. The given polynomial is:

−9y2+12y{-9y^2 + 12y}

The first step is to identify the common factors of the coefficients of the terms. The coefficients of the terms are -9 and 12. The greatest common factor of -9 and 12 is 3.

Factoring the GCF from the Polynomial

Now that we have identified the GCF, we can factor it from the polynomial. To do this, we divide each term of the polynomial by the GCF.

−9y2+12y=−3y(3y)+−3y(4){-9y^2 + 12y = -3y(3y) + -3y(4)}

Simplifying the expression, we get:

−9y2+12y=−3y(3y−4){-9y^2 + 12y = -3y(3y - 4)}

Therefore, the correct answer is:

A. −3y(3y−4){-3y(3y - 4)}

Why is this the Correct Answer?

This is the correct answer because we have factored the GCF (3) from the polynomial, and the resulting expression is a product of two binomials. The first binomial is -3y, and the second binomial is (3y - 4). This is the correct factorization of the polynomial.

Comparison with Other Options

Let's compare the correct answer with the other options:

  • B. 3(−3y2−4y){3(-3y^2 - 4y)}: This option is incorrect because the GCF is not factored from the polynomial. The GCF is 3, but it is not factored from the polynomial.
  • C. 9y(−3y2+4){9y(-3y^2 + 4)}: This option is incorrect because the GCF is not factored from the polynomial. The GCF is 3, but it is not factored from the polynomial.
  • D. y(−9y+12){y(-9y + 12)}: This option is incorrect because the GCF is not factored from the polynomial. The GCF is 3, but it is not factored from the polynomial.

Conclusion

In conclusion, factoring the GCF from a polynomial is an essential step in simplifying and solving polynomial equations. By identifying the GCF and factoring it from the polynomial, we can simplify the polynomial and make it easier to solve. In this example, we factored the GCF (3) from the polynomial −9y2+12y{-9y^2 + 12y} and obtained the correct answer: −3y(3y−4){-3y(3y - 4)}.

Common Mistakes to Avoid

When factoring the GCF from a polynomial, there are several common mistakes to avoid:

  • Not identifying the GCF: Failing to identify the GCF can lead to incorrect factorization.
  • Not factoring the GCF: Failing to factor the GCF can lead to incorrect factorization.
  • Factoring the wrong term: Factoring the wrong term can lead to incorrect factorization.

Tips for Factoring the GCF

When factoring the GCF from a polynomial, here are some tips to keep in mind:

  • Identify the GCF: Identify the GCF of the coefficients of the terms.
  • Factor the GCF: Factor the GCF from each term of the polynomial.
  • Simplify the expression: Simplify the expression by combining like terms.

Real-World Applications

Factoring the GCF from a polynomial has several real-world applications:

  • Simplifying polynomial equations: Factoring the GCF from a polynomial can simplify the polynomial and make it easier to solve.
  • Solving systems of equations: Factoring the GCF from a polynomial can help solve systems of equations.
  • Optimizing functions: Factoring the GCF from a polynomial can help optimize functions.

Conclusion

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

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

Q: How do I identify the GCF of a polynomial?

A: To identify the GCF of a polynomial, you need to identify the largest polynomial that divides each term of the polynomial without leaving a remainder. You can do this by finding the greatest common factor of the coefficients of the terms.

Q: What is the difference between the GCF and the least common multiple (LCM)?

A: The greatest common factor (GCF) is the largest polynomial that divides each term of the polynomial without leaving a remainder, while the least common multiple (LCM) is the smallest polynomial that is a multiple of each term of the polynomial.

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

A: To factor the GCF from a polynomial, you need to divide each term of the polynomial by the GCF. This will give you a product of two binomials, where the first binomial is the GCF and the second binomial is the remaining polynomial.

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

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

  • Not identifying the GCF
  • Not factoring the GCF
  • Factoring the wrong term

Q: What are some tips for factoring the GCF from a polynomial?

A: Some tips for factoring the GCF from a polynomial include:

  • Identify the GCF
  • Factor the GCF from each term of the polynomial
  • Simplify the expression by combining like terms

Q: What are some real-world applications of factoring the GCF from a polynomial?

A: Some real-world applications of factoring the GCF from a polynomial include:

  • Simplifying polynomial equations
  • Solving systems of equations
  • Optimizing functions

Q: Can you give an example of factoring the GCF from a polynomial?

A: Yes, here is an example of factoring the GCF from a polynomial:

−9y2+12y{-9y^2 + 12y}

To factor the GCF from this polynomial, we need to identify the GCF, which is 3. We can then divide each term of the polynomial by the GCF to get:

−3y(3y)+−3y(4){-3y(3y) + -3y(4)}

Simplifying the expression, we get:

−3y(3y−4){-3y(3y - 4)}

Therefore, the correct answer is:

−3y(3y−4){-3y(3y - 4)}

Q: What is the importance of factoring the GCF from a polynomial?

A: Factoring the GCF from a polynomial is an essential step in simplifying and solving polynomial equations. By identifying the GCF and factoring it from the polynomial, we can simplify the polynomial and make it easier to solve.

Q: Can you provide some additional resources for learning about factoring the GCF from a polynomial?

A: Yes, here are some additional resources for learning about factoring the GCF from a polynomial:

  • Online tutorials and videos
  • Math textbooks and workbooks
  • Online math communities and forums

Conclusion

In conclusion, factoring the GCF from a polynomial is an essential step in simplifying and solving polynomial equations. By identifying the GCF and factoring it from the polynomial, we can simplify the polynomial and make it easier to solve. We hope this Q&A article has provided you with a better understanding of factoring the GCF from a polynomial.