Question

Which polynomial function has a leading coefficient of 1, roots –2 and 7 with multiplicity 1, and root 5 with multiplicity 2? f(x) = 2(x + 7)(x + 5)(x – 2) f(x) = 2(x – 7)(x – 5)(x + 2) f(x) = (x + 7)(x + 5)(x + 5)(x – 2) f(x) = (x – 7)(x – 5)(x – 5)(x + 2)

Answer 1

From the given polynomials, we have to determine polynomial function which has a leading coefficient of 1, roots –2 and 7 with multiplicity 1, and root 5 with multiplicity 2.

1. The function f(x) = 2(x + 7)(x + 5)(x – 2) is not a function with leading coefficient as 1.

2. The function f(x) = 2(x – 7)(x – 5)(x + 2) is not a function with leading coefficient as 1.

3.The function f(x) = (x + 7)(x + 5)(x + 5)(x – 2) does'not have a root as '-2', it has a root '2' instead of '-2'.

4. The function f(x) = (x – 7)(x – 5)(x – 5)(x + 2) = $(x-7)(x-5)^{2}(x+2)$ is a function with leading coefficient as '1'.

Now, we will determine its multiplicity.

"Multiplicity denotes the total number of times a value appears in a set of variables"

So, the root '7' has multiplicity '1'.

The root '5' has multiplicity '2' as the total number of times a value appears in a variable 'x' is 2.

The root '-2' has multiplicity '1' as the total number of times a value appears in a variable 'x' is 1.

Therefore, Option 4 is the correct answer.