Which term can be used in the blank of $36 x^3-22 x^2$ __ so the greatest common factor of the resulting polynomial is $2 x$? Select two options.
2
$4 x y$
$12 x$
24
$44 y$
Answer (2)
The two terms that, when added to the polynomial 36 x 3 − 22 x 2 , produce a greatest common factor of 2 x are 4 x y and 12 x .
;
Analyze the given polynomial 36 x 3 − 22 x 2 and the options.
Determine the GCF of the resulting polynomial when each option is added.
Identify the two options that result in a GCF of 2 x .
The two terms are 4 x y and 12 x , so the answer is 4 x y , 12 x .
Explanation
Understanding the Problem We are given the polynomial 36 x 3 − 22 x 2 and asked to find two terms from the options that, when added to the polynomial, result in a greatest common factor (GCF) of 2 x .
Analyzing Each Option Let's analyze each option:
Option 1: Add 2. The polynomial becomes 36 x 3 − 22 x 2 + 2 . The GCF of the coefficients 36, -22, and 2 is 2. The terms are x 3 , x 2 , and a constant. Thus, the GCF of the polynomial is 2.
Option 2: Add 4 x y . The polynomial becomes 36 x 3 − 22 x 2 + 4 x y . The GCF of the coefficients 36, -22, and 4 is 2. Each term has a factor of x except for 4 x y which has a factor of y . The GCF of the polynomial is 2 x .
Option 3: Add 12 x . The polynomial becomes 36 x 3 − 22 x 2 + 12 x . The GCF of the coefficients 36, -22, and 12 is 2. Each term has a factor of x . Thus, the GCF of the polynomial is 2 x .
Option 4: Add 24. The polynomial becomes 36 x 3 − 22 x 2 + 24 . The GCF of the coefficients 36, -22, and 24 is 2. The terms are x 3 , x 2 , and a constant. Thus, the GCF of the polynomial is 2.
Option 5: Add 44 y . The polynomial becomes 36 x 3 − 22 x 2 + 44 y . The GCF of the coefficients 36, -22, and 44 is 2. The terms are x 3 , x 2 , and y . Thus, the GCF of the polynomial is 2.
Identifying Correct Options From the analysis above, the two options that result in a GCF of 2 x are 4 x y and 12 x .
Final Answer Therefore, the two terms that can be used in the blank so that the greatest common factor of the resulting polynomial is 2 x are 4 x y and 12 x .
Examples
Understanding greatest common factors is useful in simplifying expressions and solving equations. For example, if you are designing a rectangular garden and want to enclose it with fencing, knowing the GCF of the garden's dimensions can help you determine the largest possible equal-length sections you can use for the fence posts, minimizing waste and ensuring even spacing. This concept is also used in cryptography and data compression to optimize algorithms and reduce storage space.
