Answer :
- Find the greatest common factor (GCF) of the coefficients: The GCF of 6, 45, 15, and 21 is 3.
- Find the greatest common factor of the variable terms: The GCF of $x^{12}, x^7, x^6,$ and $x^4$ is $x^4$.
- Multiply the GCF of the coefficients and the GCF of the variable terms to find the overall GCF: $3x^4$.
- Factor out the overall GCF from each term in the expression: $6 x^{12}+45 x^7+15 x^6+21 x^4 = 3x^4(2x^8 + 15x^3 + 5x^2 + 7)$.
- The final answer is $\boxed{3x^4(2x^8 + 15x^3 + 5x^2 + 7)}$.
### Explanation
1. Understanding the Problem
We are asked to rewrite the expression $6 x^{12}+45 x^7+15 x^6+21 x^4$ by factoring out the greatest common factor (GCF). This involves finding the largest factor that divides all terms in the expression. We will find the GCF of the coefficients and the GCF of the variable terms separately, and then combine them to find the overall GCF.
2. Finding the GCF of the Coefficients
First, let's find the greatest common factor (GCF) of the coefficients: 6, 45, 15, and 21. We can use the Euclidean algorithm or prime factorization to find the GCF. The prime factorization of each number is:
$6 = 2 \times 3$
$45 = 3^2 \times 5$
$15 = 3 \times 5$
$21 = 3 \times 7$
The only common prime factor among all these numbers is 3, and it appears with the lowest power of 1. Therefore, the GCF of the coefficients is 3.
3. Finding the GCF of the Variable Terms
Next, let's find the greatest common factor of the variable terms: $x^{12}, x^7, x^6,$ and $x^4$. The GCF of variable terms is the lowest power of x present in all terms. In this case, the lowest power of x is $x^4$. Therefore, the GCF of the variable terms is $x^4$.
4. Finding the Overall GCF
Now, we multiply the GCF of the coefficients and the GCF of the variable terms to find the overall GCF of the expression. The GCF of the coefficients is 3, and the GCF of the variable terms is $x^4$. Therefore, the overall GCF of the expression is $3x^4$.
5. Factoring out the GCF
We factor out the overall GCF, $3x^4$, from each term in the expression:
$6x^{12} = 3x^4(2x^8)$
$45x^7 = 3x^4(15x^3)$
$15x^6 = 3x^4(5x^2)$
$21x^4 = 3x^4(7)$
So, we can rewrite the expression as:
$6 x^{12}+45 x^7+15 x^6+21 x^4 = 3x^4(2x^8 + 15x^3 + 5x^2 + 7)$
6. Final Answer
Therefore, the expression rewritten by taking out the greatest common factor is $3x^4(2x^8 + 15x^3 + 5x^2 + 7)$.
### Examples
Factoring out the greatest common factor is a fundamental technique in algebra that simplifies expressions and helps in solving equations. For example, if you are designing a rectangular garden and want to express the area in terms of its dimensions, factoring can help you find the simplest form of the expression. Similarly, in physics, when dealing with polynomial equations describing motion or energy, factoring simplifies the equations, making them easier to analyze and solve. This technique is also used in computer science for optimizing code and reducing computational complexity.
- Find the greatest common factor of the variable terms: The GCF of $x^{12}, x^7, x^6,$ and $x^4$ is $x^4$.
- Multiply the GCF of the coefficients and the GCF of the variable terms to find the overall GCF: $3x^4$.
- Factor out the overall GCF from each term in the expression: $6 x^{12}+45 x^7+15 x^6+21 x^4 = 3x^4(2x^8 + 15x^3 + 5x^2 + 7)$.
- The final answer is $\boxed{3x^4(2x^8 + 15x^3 + 5x^2 + 7)}$.
### Explanation
1. Understanding the Problem
We are asked to rewrite the expression $6 x^{12}+45 x^7+15 x^6+21 x^4$ by factoring out the greatest common factor (GCF). This involves finding the largest factor that divides all terms in the expression. We will find the GCF of the coefficients and the GCF of the variable terms separately, and then combine them to find the overall GCF.
2. Finding the GCF of the Coefficients
First, let's find the greatest common factor (GCF) of the coefficients: 6, 45, 15, and 21. We can use the Euclidean algorithm or prime factorization to find the GCF. The prime factorization of each number is:
$6 = 2 \times 3$
$45 = 3^2 \times 5$
$15 = 3 \times 5$
$21 = 3 \times 7$
The only common prime factor among all these numbers is 3, and it appears with the lowest power of 1. Therefore, the GCF of the coefficients is 3.
3. Finding the GCF of the Variable Terms
Next, let's find the greatest common factor of the variable terms: $x^{12}, x^7, x^6,$ and $x^4$. The GCF of variable terms is the lowest power of x present in all terms. In this case, the lowest power of x is $x^4$. Therefore, the GCF of the variable terms is $x^4$.
4. Finding the Overall GCF
Now, we multiply the GCF of the coefficients and the GCF of the variable terms to find the overall GCF of the expression. The GCF of the coefficients is 3, and the GCF of the variable terms is $x^4$. Therefore, the overall GCF of the expression is $3x^4$.
5. Factoring out the GCF
We factor out the overall GCF, $3x^4$, from each term in the expression:
$6x^{12} = 3x^4(2x^8)$
$45x^7 = 3x^4(15x^3)$
$15x^6 = 3x^4(5x^2)$
$21x^4 = 3x^4(7)$
So, we can rewrite the expression as:
$6 x^{12}+45 x^7+15 x^6+21 x^4 = 3x^4(2x^8 + 15x^3 + 5x^2 + 7)$
6. Final Answer
Therefore, the expression rewritten by taking out the greatest common factor is $3x^4(2x^8 + 15x^3 + 5x^2 + 7)$.
### Examples
Factoring out the greatest common factor is a fundamental technique in algebra that simplifies expressions and helps in solving equations. For example, if you are designing a rectangular garden and want to express the area in terms of its dimensions, factoring can help you find the simplest form of the expression. Similarly, in physics, when dealing with polynomial equations describing motion or energy, factoring simplifies the equations, making them easier to analyze and solve. This technique is also used in computer science for optimizing code and reducing computational complexity.
