What is the greatest common factor of the polynomial below?
[tex]$18 x^3-6 x$[/tex]
A. [tex]$6 x^2$[/tex]
B. [tex]$3 x$[/tex]
C. [tex]$3 x^2$[/tex]
D. [tex]$6 x$[/tex]
Answer (1)
The greatest common factor of the polynomial 18 x 3 − 6 x is found by identifying the GCF of the coefficients and the variables separately.
The GCF of the coefficients 18 and 6 is 6.
The GCF of the variable terms x 3 and x is x .
Combining these, the GCF of the polynomial is 6 x .
Therefore, the answer is 6 x .
Explanation
Understanding the Problem We are asked to find the greatest common factor (GCF) of the polynomial 18 x 3 − 6 x . The GCF is the largest expression that divides evenly into each term of the polynomial.
Finding the GCF of the Coefficients First, let's find the GCF of the coefficients, 18 and 6. The factors of 18 are 1, 2, 3, 6, 9, and 18. The factors of 6 are 1, 2, 3, and 6. The greatest common factor of 18 and 6 is 6.
Finding the GCF of the Variables Next, let's find the GCF of the variable terms, x 3 and x . The GCF is the lowest power of x that appears in both terms, which is x 1 or simply x .
Combining the GCFs Now, we combine the GCF of the coefficients and the GCF of the variables to find the GCF of the entire polynomial. The GCF is 6 x .
Final Answer Therefore, the greatest common factor of the polynomial 18 x 3 − 6 x is 6 x . Looking at the options, we see that option D is 6 x .
Examples
Understanding the greatest common factor is useful in many real-world situations. For example, if you are tiling a rectangular floor and want to use the largest possible square tiles without cutting any tiles, you would find the GCF of the length and width of the floor. Similarly, in manufacturing, if you need to cut two different lengths of material into equal pieces with no waste, the GCF helps determine the maximum length of each piece. This concept also applies in simplifying fractions and algebraic expressions, making complex problems easier to manage.
