1. Use the Intermediate Value Theorem to show that the polynomial function has a zero in the given interval.

f(x) = 13x^4 - 5x^2 +7x -1; [3,0]
Enter the value of (-3).

2. Use the Intermediate Value Theorem to show that the polynomial
function has a zero in the given interval.
f(x) = 2x^3 + 8x^2 - 7x + 9; [-9, -4]
Find the value of (-9)

1. since f(0) = -1 and f(3) = 1028, there must be some value 0 < c < 3 where f(c) = 0.

Dunno what you want with -3.

2. since f(-9) = -738 and f(-4) = 37, there is some -9 < c < -4 where f(c) = 0.

To use the Intermediate Value Theorem to show that a polynomial function has a zero in a given interval, we need to evaluate the function at the endpoints of the interval and check if the signs of the function values at the endpoints are different. If they are different, then the function must have a zero (or a root) somewhere within the interval.

1. For the first question:
We have the polynomial function f(x) = 13x^4 - 5x^2 + 7x - 1

To find the value of f(-3), we substitute x = -3 into the function:
f(-3) = 13(-3)^4 - 5(-3)^2 + 7(-3) - 1
= 13(81) - 5(9) - 21 - 1
= 1053 - 45 - 21 - 1
= 986

Since the interval is [3, 0], we need to find the values of f(3) and f(0) as well.

To find the value of f(3), we substitute x = 3 into the function:
f(3) = 13(3)^4 - 5(3)^2 + 7(3) - 1
= 13(81) - 5(9) + 21 - 1
= 1053 - 45 + 21 - 1
= 1028

To find the value of f(0), we substitute x = 0 into the function:
f(0) = 13(0)^4 - 5(0)^2 + 7(0) - 1
= 0 - 0 + 0 - 1
= -1

Now, we check the signs of these values:
f(-3) = 986 (positive)
f(3) = 1028 (positive)
f(0) = -1 (negative)

Since f(-3) and f(0) have different signs, we can conclude that the polynomial function f(x) = 13x^4 - 5x^2 + 7x - 1 has a zero (or a root) in the interval [3, 0].

2. For the second question:
We have the polynomial function f(x) = 2x^3 + 8x^2 - 7x + 9

To find the value of f(-9), we substitute x = -9 into the function:
f(-9) = 2(-9)^3 + 8(-9)^2 - 7(-9) + 9
= 2(-729) + 8(81) + 63 + 9
= -1458 + 648 + 63 + 9
= -738

Since the interval is [-9, -4], we need to find the values of f(-9) and f(-4) as well.

To find the value of f(-4), we substitute x = -4 into the function:
f(-4) = 2(-4)^3 + 8(-4)^2 - 7(-4) + 9
= 2(-64) + 8(16) + 28 + 9
= -128 + 128 + 28 + 9
= 37

Now, we check the signs of these values:
f(-9) = -738 (negative)
f(-4) = 37 (positive)

Since f(-9) and f(-4) have different signs, we can conclude that the polynomial function f(x) = 2x^3 + 8x^2 - 7x + 9 has a zero (or a root) in the interval [-9, -4].