Factor Out The Greatest Common Factor (GCF) From The Expression: 64 D 5 − 24 D 2 64d^5 - 24d^2 64 D 5 − 24 D 2 Options:A. 8 D 2 ⋅ ( 8 D 3 − 3 8d^2 \cdot (8d^3 - 3 8 D 2 ⋅ ( 8 D 3 − 3 ]B. 8 D ⋅ ( 8 D 4 − 3 D 8d \cdot (8d^4 - 3d 8 D ⋅ ( 8 D 4 − 3 D ]C. 4 D ⋅ ( 16 D 4 − 6 D 4d \cdot (16d^4 - 6d 4 D ⋅ ( 16 D 4 − 6 D ]D. 8 D 2 ⋅ ( 8 D 3 − 3 8d^2 \cdot (8d^3 - 3 8 D 2 ⋅ ( 8 D 3 − 3 ]

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Factor out the Greatest Common Factor (GCF) from the expression: 64d5−24d264d^5 - 24d^2

Understanding the Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is the largest expression that divides each term of a polynomial without leaving a remainder. In other words, it is the product of the common factors of the terms in the polynomial. Factoring out the GCF is an essential step in simplifying polynomials and solving equations.

Identifying the GCF

To factor out the GCF from the expression 64d5−24d264d^5 - 24d^2, we need to identify the common factors of the two terms. The first step is to find the greatest power of each variable that divides both terms. In this case, the greatest power of dd that divides both terms is d2d^2. The coefficients of the two terms are 64 and 24, respectively. The greatest common factor of these coefficients is 8.

Factoring out the GCF

Now that we have identified the GCF, we can factor it out from the expression. To do this, we divide each term by the GCF and write the result as a product of the GCF and a new expression. In this case, we divide each term by 8d^2:

64d58d2=8d3\frac{64d^5}{8d^2} = 8d^3

−24d28d2=−3\frac{-24d^2}{8d^2} = -3

Therefore, the expression 64d5−24d264d^5 - 24d^2 can be factored as:

8d2⋅(8d3−3)8d^2 \cdot (8d^3 - 3)

Evaluating the Options

Now that we have factored out the GCF, we can evaluate the options:

A. 8d⋅(8d4−3d)8d \cdot (8d^4 - 3d)

B. 8d⋅(8d4−3d)8d \cdot (8d^4 - 3d)

C. 4d⋅(16d4−6d)4d \cdot (16d^4 - 6d)

D. 8d2⋅(8d3−3)8d^2 \cdot (8d^3 - 3)

Only option D matches the factored expression we obtained earlier.

Conclusion

In this article, we learned how to factor out the Greatest Common Factor (GCF) from a polynomial expression. We identified the GCF by finding the greatest power of each variable that divides both terms and the greatest common factor of the coefficients. We then factored out the GCF by dividing each term by the GCF and writing the result as a product of the GCF and a new expression. We evaluated the options and found that only option D matches the factored expression.

Greatest Common Factor (GCF) Examples

Here are some examples of factoring out the GCF from polynomial expressions:

  • 12x3−18x212x^3 - 18x^2: The GCF is 6x^2. Factoring out the GCF gives: 6x2⋅(2x−3)6x^2 \cdot (2x - 3)
  • 20y4−15y320y^4 - 15y^3: The GCF is 5y^3. Factoring out the GCF gives: 5y3⋅(4y−3)5y^3 \cdot (4y - 3)
  • 36z5−24z436z^5 - 24z^4: The GCF is 12z^4. Factoring out the GCF gives: 12z4⋅(3z−2)12z^4 \cdot (3z - 2)

Greatest Common Factor (GCF) Properties

Here are some properties of the Greatest Common Factor (GCF):

  • The GCF of a set of numbers is the product of the common factors of the numbers.
  • The GCF of a set of numbers is always less than or equal to the smallest number in the set.
  • The GCF of a set of numbers is always greater than or equal to the greatest common divisor of the numbers.
  • The GCF of a set of numbers can be found by listing the factors of each number and finding the common factors.

Greatest Common Factor (GCF) Applications

The Greatest Common Factor (GCF) has many applications in mathematics and other fields. Here are a few examples:

  • Simplifying polynomials: Factoring out the GCF is an essential step in simplifying polynomials and solving equations.
  • Finding the greatest common divisor: The GCF can be used to find the greatest common divisor of a set of numbers.
  • Solving equations: The GCF can be used to solve equations by factoring out the GCF and then solving for the remaining expression.
  • Cryptography: The GCF is used in cryptography to encrypt and decrypt messages.

Greatest Common Factor (GCF) Conclusion

In conclusion, the Greatest Common Factor (GCF) is an essential concept in mathematics that has many applications. It is used to simplify polynomials, find the greatest common divisor, solve equations, and encrypt and decrypt messages. We learned how to factor out the GCF from a polynomial expression and evaluated the options. We also discussed the properties and applications of the GCF.
Greatest Common Factor (GCF) Q&A

Q: What is the Greatest Common Factor (GCF)?

A: The Greatest Common Factor (GCF) is the largest expression that divides each term of a polynomial without leaving a remainder. It is the product of the common factors of the terms in the polynomial.

Q: How do I find the GCF of a set of numbers?

A: To find the GCF of a set of numbers, you can list the factors of each number and find the common factors. You can also use the following steps:

  1. Find the prime factorization of each number.
  2. Identify the common prime factors.
  3. Multiply the common prime factors together.

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

A: The GCF is the largest expression that divides each term of a polynomial without leaving a remainder, while the LCM is the smallest expression that is a multiple of each term of a polynomial. The GCF and LCM are related by the following equation:

GCF(a, b) × LCM(a, b) = a × b

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

A: To factor out the GCF from a polynomial expression, you can follow these steps:

  1. Identify the GCF of the terms in the polynomial.
  2. Divide each term by the GCF.
  3. Write the result as a product of the GCF and a new expression.

Q: What are some common mistakes to avoid when finding the GCF?

A: Some common mistakes to avoid when finding the GCF include:

  • Not listing all the factors of each number.
  • Not identifying the common factors.
  • Not multiplying the common factors together.
  • Not checking for common factors in the prime factorization.

Q: How do I use the GCF to simplify a polynomial expression?

A: To use the GCF to simplify a polynomial expression, you can follow these steps:

  1. Factor out the GCF from the polynomial expression.
  2. Simplify the resulting expression.
  3. Write the final expression in factored form.

Q: What are some real-world applications of the GCF?

A: Some real-world applications of the GCF include:

  • Simplifying polynomials in algebra.
  • Finding the greatest common divisor in number theory.
  • Solving equations in calculus.
  • Encrypting and decrypting messages in cryptography.

Q: Can you give me some examples of finding the GCF?

A: Here are some examples of finding the GCF:

  • Find the GCF of 12 and 18: The GCF is 6.
  • Find the GCF of 24 and 30: The GCF is 6.
  • Find the GCF of 36 and 48: The GCF is 12.

Q: Can you give me some examples of factoring out the GCF?

A: Here are some examples of factoring out the GCF:

  • Factor out the GCF from 12x^3 - 18x^2: The GCF is 6x^2. Factoring out the GCF gives: 6x^2(2x - 3)
  • Factor out the GCF from 20y^4 - 15y^3: The GCF is 5y^3. Factoring out the GCF gives: 5y^3(4y - 3)
  • Factor out the GCF from 36z^5 - 24z^4: The GCF is 12z^4. Factoring out the GCF gives: 12z^4(3z - 2)

Q: Can you give me some examples of using the GCF to simplify a polynomial expression?

A: Here are some examples of using the GCF to simplify a polynomial expression:

  • Simplify the polynomial expression 12x^3 - 18x^2: The GCF is 6x^2. Factoring out the GCF gives: 6x^2(2x - 3)
  • Simplify the polynomial expression 20y^4 - 15y^3: The GCF is 5y^3. Factoring out the GCF gives: 5y^3(4y - 3)
  • Simplify the polynomial expression 36z^5 - 24z^4: The GCF is 12z^4. Factoring out the GCF gives: 12z^4(3z - 2)