Question

Find the Highest Common Factor (HCF) of the following polynomials: x² - 12x - 28 + 16y - y², x² + 2xy² + 2y, x² - y² + 4y - 4.

A) (x - 2)(x - 14)
B) (x + 2y + 2)
C) (x - 2)(x + 2)
D) (x + 2)(x + 14)

Answer 1

The Highest Common Factor (HCF) of the given polynomials is C: (x - 2)(x + 2), which is the factor present in all the given polynomials.

Step-by-step explanation:

The student is asked to find the Highest Common Factor (HCF) of the given polynomials. To find the HCF, we must look for the common polynomial factor that is present in all of the given polynomials. Checking the provided options against each polynomial will allow us to determine which is the correct HCF.

First Polynomial: x² - 12x - 28 + 16y - y²
Second Polynomial: x² + 2xy² + 2y
Third Polynomial: x² - y² + 4y - 4

To quickly find the HCF, we can look at the GCD of individual terms. The x² term is present in all the polynomials, so the HCF will at least contain x. Additionally, the difference of squares is a common factor we can look for - in this case, x² - y², which is (x - y)(x + y).

After looking at these patterns and choices, we find that the correct answer is C: (x - 2)(x + 2), which is the factor of x² - y² and is a factor common to all three given polynomials.