Use the intermediate value theorem for polynomials to show that the polynomial function has a real zero between the numbers given.
-$f ( x ) = 9 x ^ { 4 } - 8 x ^ { 2 } - 7 ; 1$ and 2
A) $f ( 1 ) = 6$ and $f ( 2 ) = - 105$
B) $f ( 1 ) = - 5$ and $f ( 2 ) = 106$
C) $f ( 1 ) = 6$ and $f ( 2 ) = 105$
D) $f ( 1 ) = - 6$ and $f ( 2 ) = 105$

Question

Quizplus · Accepted Answer

To show that a polynomial function has a real zero between two numbers using the intermediate value theorem, we need to calculate the value of the function at these two numbers and show that the function values have opposite signs. For $f(x) = 9x^4 - 8x^2 - 7$, calculating $f(1)$ and $f(2)$ gives: $f(1) = 9(1)^4 - 8(1)^2 - 7 = 9 - 8 - 7 = -6$ and $f(2) = 9(2)^4 - 8(2)^2 - 7 = 9(16) - 8(4) - 7 = 144 - 32 - 7 = 105$. Since $f(1) = -6$ and $f(2) = 105$, and these values have opposite signs, there must be at least one real zero between 1 and 2 according to the intermediate value theorem.

Use the Intermediate Value Theorem for Polynomials to Show That f(x)=9x4−8x2−7;1f ( x ) = 9 x ^ { 4 } - 8 x ^ { 2 } - 7 ; 1f(x)=9x4−8x2−7;1

Use the Intermediate Value Theorem for Polynomials to Show That $f ( x ) = 9 x ^ { 4 } - 8 x ^ { 2 } - 7 ; 1$