What is the greatest common factor of the polynomial below?
[tex]12 x^2-9 x[/tex]
A. [tex]3 x[/tex]
B. [tex]4 x[/tex]
C. [tex]3 x^2[/tex]
D. [tex]4 x^2[/tex]
Answer (1)
Find the greatest common factor of the coefficients 12 and 9, which is 3.
Find the greatest common factor of the variable parts x 2 and x , which is x .
Combine the GCF of the coefficients and variables to get 3 x .
The greatest common factor of the polynomial 12 x 2 − 9 x is 3 x .
Explanation
Understanding the Problem We are asked to find the greatest common factor (GCF) of the polynomial 12 x 2 − 9 x . The GCF is the largest expression that divides evenly into both terms of the polynomial.
Finding GCF of Coefficients First, let's find the GCF of the coefficients, 12 and 9. The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 9 are 1, 3, and 9. The greatest common factor of 12 and 9 is 3.
Finding GCF of Variables Now, let's look at the variable part of each term. The first term has x 2 and the second term has x . The common variable factor is x (since x divides both x 2 and x ).
Combining GCFs Combining the GCF of the coefficients and the GCF of the variables, the greatest common factor of 12 x 2 and − 9 x is 3 x .
Final Answer Therefore, the greatest common factor of the polynomial 12 x 2 − 9 x is 3 x .
Examples
Understanding the greatest common factor is useful in many real-life situations. For example, suppose you want to tile a rectangular floor that is 12 feet wide and 9 feet long using square tiles. The greatest common factor of 12 and 9, which is 3, tells you that the largest square tile you can use is 3 feet by 3 feet. This ensures that you can tile the floor completely without cutting any tiles. Similarly, GCF helps in simplifying fractions, dividing quantities evenly, and optimizing resource allocation.
