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Answer :
- Find the GCF of the coefficients: The GCF of 48, 6, and 26 is 2.
- Find the smallest power of x: The smallest power of x in the terms $48x^6$, $6x^2$, and $-26x^3$ is $x^2$.
- Combine the GCF of the coefficients and the smallest power of x: The GCF of the polynomial is $2x^2$.
- The greatest common factor of the polynomial $48 x^6+6 x^2-26 x^3$ is $\boxed{2x^2}$.
### Explanation
1. Understanding the Problem
We are asked to find the greatest common factor (GCF) of the polynomial $48 x^6+6 x^2-26 x^3$. The GCF is the largest expression that divides evenly into each term of the polynomial.
2. Finding the GCF of the Coefficients
First, let's find the GCF of the coefficients: 48, 6, and -26. The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. The factors of 6 are 1, 2, 3, and 6. The factors of 26 are 1, 2, 13, and 26. The greatest common factor of 48, 6, and 26 is 2.
3. Finding the Smallest Power of x
Next, let's find the smallest power of $x$ that appears in all terms. The terms are $48x^6$, $6x^2$, and $-26x^3$. The powers of $x$ are 6, 2, and 3. The smallest of these is 2, so the GCF will contain $x^2$.
4. Combining the GCFs
Therefore, the GCF of the polynomial $48 x^6+6 x^2-26 x^3$ is $2x^2$.
5. Final Answer
In conclusion, the greatest common factor of the given polynomial is $2x^2$.
### Examples
Understanding the GCF is useful in simplifying algebraic expressions and solving equations. For example, if you are trying to factor the expression $48 x^6+6 x^2-26 x^3$, finding the GCF allows you to write it as $2x^2(24x^4 + 3 - 13x)$, which can make further analysis or simplification easier. This technique is also used in various engineering and physics problems where simplifying complex expressions is necessary to find solutions.
- Find the smallest power of x: The smallest power of x in the terms $48x^6$, $6x^2$, and $-26x^3$ is $x^2$.
- Combine the GCF of the coefficients and the smallest power of x: The GCF of the polynomial is $2x^2$.
- The greatest common factor of the polynomial $48 x^6+6 x^2-26 x^3$ is $\boxed{2x^2}$.
### Explanation
1. Understanding the Problem
We are asked to find the greatest common factor (GCF) of the polynomial $48 x^6+6 x^2-26 x^3$. The GCF is the largest expression that divides evenly into each term of the polynomial.
2. Finding the GCF of the Coefficients
First, let's find the GCF of the coefficients: 48, 6, and -26. The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. The factors of 6 are 1, 2, 3, and 6. The factors of 26 are 1, 2, 13, and 26. The greatest common factor of 48, 6, and 26 is 2.
3. Finding the Smallest Power of x
Next, let's find the smallest power of $x$ that appears in all terms. The terms are $48x^6$, $6x^2$, and $-26x^3$. The powers of $x$ are 6, 2, and 3. The smallest of these is 2, so the GCF will contain $x^2$.
4. Combining the GCFs
Therefore, the GCF of the polynomial $48 x^6+6 x^2-26 x^3$ is $2x^2$.
5. Final Answer
In conclusion, the greatest common factor of the given polynomial is $2x^2$.
### Examples
Understanding the GCF is useful in simplifying algebraic expressions and solving equations. For example, if you are trying to factor the expression $48 x^6+6 x^2-26 x^3$, finding the GCF allows you to write it as $2x^2(24x^4 + 3 - 13x)$, which can make further analysis or simplification easier. This technique is also used in various engineering and physics problems where simplifying complex expressions is necessary to find solutions.
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