What is the greatest common factor of the terms in the polynomial [tex]$4 x^4-32 x^3-60 x^2$[/tex]?
Answer (2)
Find the greatest common factor (GCF) of the coefficients 4, -32, and -60, which is 4.
Find the GCF of the variable parts x 4 , x 3 , and x 2 , which is x 2 .
Multiply the GCF of the coefficients and the GCF of the variable parts: 4 × x 2 = 4 x 2 .
The greatest common factor of the polynomial 4 x 4 − 32 x 3 − 60 x 2 is 4 x 2 .
Explanation
Understanding the Problem We are asked to find the greatest common factor (GCF) of the terms in the polynomial 4 x 4 − 32 x 3 − 60 x 2 . The GCF is the largest factor that divides all terms of the polynomial.
Finding the GCF of the Coefficients First, let's find the GCF of the coefficients: 4, -32, and -60. The GCF of these numbers is 4, since 4 divides all three numbers. We can write the coefficients as 4 = 4 × 1 , − 32 = 4 × − 8 , and − 60 = 4 × − 15 .
Finding the GCF of the Variable Parts Next, let's find the GCF of the variable parts: x 4 , x 3 , and x 2 . The GCF of these terms is x 2 , since it is the lowest power of x present in all terms. We can write the variable parts as x 4 = x 2 × x 2 , x 3 = x 2 × x , and x 2 = x 2 × 1 .
Combining the GCFs Now, we multiply the GCF of the coefficients and the GCF of the variable parts to find the GCF of the entire polynomial. The GCF is 4 × x 2 = 4 x 2 .
Final Answer Therefore, the greatest common factor of the terms in the polynomial 4 x 4 − 32 x 3 − 60 x 2 is 4 x 2 .
Examples
Understanding the greatest common factor is useful in many real-life 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, if you are packaging items into boxes and want to ensure each box has the same number of each item, the GCF helps determine the maximum number of boxes you can create. Factoring polynomials, like finding the GCF, is a fundamental skill that simplifies many mathematical and practical problems.
The greatest common factor of the polynomial 4 x 4 − 32 x 3 − 60 x 2 is 4 x 2 , found by determining the GCF of the coefficients and the variable parts separately and then combining them. The GCF of the coefficients 4, -32, and -60 is 4, and the GCF of the variable parts x 4 , x 3 , x 2 is x 2 . Multiplying these together gives us the final result of 4 x 2 .
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