Questions: (a) Determine if the upper bound theorem identifies 2 as an upper bound for the real zeros of f(x). (b) Determine if the lower bound theorem identifies -7 as a lower bound for the real zeros of f(x). f(x)=x^5+5x^4+3x^3+51x^2-94x+38

Solution Steps

(a) To determine if 2 is an upper bound for the real zeros of \( f(x) \), we can use the Upper Bound Theorem. According to this theorem, if we perform synthetic division of \( f(x) \) by \( x - c \) (where \( c \) is the potential upper bound) and all the coefficients of the resulting polynomial are non-negative, then \( c \) is an upper bound for the real zeros of \( f(x) \).

(b) To determine if -7 is a lower bound for the real zeros of \( f(x) \), we can use the Lower Bound Theorem. According to this theorem, if we perform synthetic division of \( f(x) \) by \( x - c \) (where \( c \) is the potential lower bound) and the coefficients of the resulting polynomial alternate in sign, then \( c \) is a lower bound for the real zeros of \( f(x) \).

To determine if \( 2 \) is an upper bound for the real zeros of \( f(x) = x^{5} + 5x^{4} + 3x^{3} + 51x^{2} - 94x + 38 \), we performed synthetic division. The resulting coefficients were:

\[ [1, 7, 17, 85, 76, 190] \]

Since all coefficients are non-negative, we conclude that \( 2 \) is indeed an upper bound for the real zeros of \( f(x) \).

Next, we checked if \( -7 \) is a lower bound for the real zeros of \( f(x) \). The synthetic division yielded the coefficients:

\[ [1, -2, 17, -68, 382, -2636] \]

The coefficients alternate in sign, confirming that \( -7 \) is a lower bound for the real zeros of \( f(x) \).

Final Answer