1.is the function f(X)=4-7x^5 a polynomial function? if so state its degree and leading coefficient.

6.use the remainder theorem to determine if x-2 is a factor of the polynomial f(x)=3x^5-7x^3-11x^2+2

1. It is a polynomial function. The degree is 5 and the leading coefficient is -7

The whole quiz?

1. Yes, the function f(x) = 4 - 7x^5 is a polynomial function. Its degree is 5 and the leading coefficient is -7.

6. To determine if x - 2 is a factor of f(x) = 3x^5 - 7x^3 - 11x^2 + 2, we can use the remainder theorem. First, substitute x = 2 into f(x):

f(2) = 3(2)^5 - 7(2)^3 - 11(2)^2 + 2
= 3(32) - 7(8) - 11(4) + 2
= 96 - 56 - 44 + 2
= -2

Since the remainder is not zero (it is equal to -2), x - 2 is not a factor of the polynomial f(x).

1. Yes, the function f(X) = 4-7x^5 is a polynomial function because it is in the form ax^n, where a is a constant and n is a non-negative integer.

The degree of the polynomial is determined by the highest power of x in the function, which in this case is 5. Therefore, the degree of the function f(X) = 4-7x^5 is 5.

The leading coefficient is the coefficient of the term with the highest power of x, which is -7 in this case. Therefore, the leading coefficient of the function f(X) = 4-7x^5 is -7.

6. To determine if x-2 is a factor of the polynomial f(x) = 3x^5-7x^3-11x^2+2 using the remainder theorem, we need to check if f(2) equals zero.

First, substitute x = 2 into the polynomial:

f(2) = 3(2)^5 - 7(2)^3 - 11(2)^2 + 2
= 3(32) - 7(8) - 11(4) + 2
= 96 - 56 - 44 + 2
= -2

Since f(2) is not equal to zero, x-2 is not a factor of the polynomial f(x) = 3x^5-7x^3-11x^2+2.

To determine if a function is a polynomial function, we need to check if all the terms have non-negative integer exponents and if there are no variables in the denominators or inside radicals.

1. For the function f(X) = 4 - 7X^5:
- All the exponents (5 in this case) of the variable X are non-negative integers, so it satisfies the first condition.
- There are no variables in the denominators or inside radicals, so it satisfies the second condition.

Therefore, the function f(X) = 4 - 7X^5 is indeed a polynomial function.

To find the degree and leading coefficient:
- The degree of a polynomial is determined by the highest exponent of the variable. In this case, the highest exponent of X is 5, so the degree of the polynomial is 5.
- The leading coefficient is the coefficient of the term with the highest exponent. In this case, the coefficient is -7.

Therefore, the degree of the polynomial is 5 and the leading coefficient is -7.

2. To use the remainder theorem to determine if (x - 2) is a factor of the polynomial f(x) = 3x^5 - 7x^3 - 11x^2 + 2, we follow these steps:
- Set up the synthetic division by writing the coefficients of the terms in descending order: 3, 0, -7, -11, 0, 2.
- Use the divisor as the value to divide into the polynomial, which is 2.
- Apply synthetic division to get the remainder.

Here's the step-by-step process:
--------------
2 | 3 0 -7 -11 0 2
6 12 10 -2 4
--------------
6 5 -1 4 6

The remainder obtained is 6.

The remainder theorem states that if the remainder obtained from synthetic division is zero, then the divisor is a factor of the polynomial.

In this case, the remainder is 6, so (x - 2) is not a factor of the polynomial f(x) = 3x^5 - 7x^3 - 11x^2 + 2.