Factor out the GCF of the polynomial [tex]$2 x^6+2 x^5$[/tex].
Answer (2)
Find the GCF of the coefficients: 2.
Find the GCF of the variable terms: x 5 .
Multiply the GCFs: 2 x 5 .
Factor out the GCF: 2 x 5 ( x + 1 ) . The answer is 2 x 5 ( x + 1 ) .
Explanation
Understanding the Problem We are asked to factor out the greatest common factor (GCF) from the polynomial 2 x 6 + 2 x 5 . The GCF is the largest expression that divides evenly into both terms of the polynomial.
Finding the GCF of the Coefficients First, we find the GCF of the coefficients. The coefficients are 2 and 2. The greatest common factor of 2 and 2 is 2.
Finding the GCF of the Variable Terms Next, we find the GCF of the variable terms. The variable terms are x 6 and x 5 . The greatest common factor of x 6 and x 5 is x 5 because x 5 is the highest power of x that divides both x 6 and x 5 .
Finding the GCF of the Polynomial Now, we multiply the GCF of the coefficients and the GCF of the variable terms to find the GCF of the polynomial. The GCF of the polynomial is 2 × x 5 = 2 x 5 .
Factoring out the GCF Finally, we factor out the GCF from the polynomial: 2 x 6 + 2 x 5 = 2 x 5 ( x + 1 ) .
Selecting the Correct Option Comparing our result with the given options, we see that the correct answer is 2 x 5 ( x + 1 ) .
Examples
Factoring out the GCF is a fundamental skill in algebra. For example, suppose you want to simplify the expression 4 x 5 2 x 6 + 2 x 5 . By factoring out the GCF, 2 x 5 , from the numerator, you get 4 x 5 2 x 5 ( x + 1 ) . Then you can cancel the common factor of 2 x 5 from the numerator and denominator, which simplifies the expression to 2 x + 1 . This technique is useful in simplifying complex expressions and solving equations.
To factor the polynomial 2 x 6 + 2 x 5 , we find the GCF of the coefficients is 2 and the GCF of the variable terms is x 5 . This leads us to factor the polynomial into 2 x 5 ( x + 1 ) .
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