Question(a) Determine if the upper bound theorem identifies 5 as an upper bound for the real zeros of $f(x)$ . (b) Determine if the lower bound theorem identifies -5 as a lower bound for the real zeros of $f(x)$ . $f(x)=x^{4}+14 x^{3}+x^{2}+4 x+39$

Studdy Solution

STEP 1

1. We are using the Upper Bound Theorem and Lower Bound Theorem for polynomials.
2. The Upper Bound Theorem states that if you perform synthetic division on a polynomial $f(x)$ by $x - c$ and all the coefficients in the result are non-negative, then $c$ is an upper bound for the real zeros of $f(x)$ .
3. The Lower Bound Theorem states that if you perform synthetic division on a polynomial $f(x)$ by $x - c$ and the coefficients in the result alternate in sign, then $c$ is a lower bound for the real zeros of $f(x)$ .

STEP 2

1. Use synthetic division to test if 5 is an upper bound for the real zeros of $f(x)$ .
2. Use synthetic division to test if -5 is a lower bound for the real zeros of $f(x)$ .

STEP 3

Perform synthetic division of $f(x) = x^{4} + 14x^{3} + x^{2} + 4x + 39$ by $x - 5$ .
Write down the coefficients of $f(x)$ : $1, 14, 1, 4, 39$ .
Perform synthetic division:
$\begin{array}{r|rrrrr} 5 & 1 & 14 & 1 & 4 & 39 \\ & & 5 & 95 & 480 & 2420 \\ \hline & 1 & 19 & 96 & 484 & 2459 \\ \end{array}$
The row at the bottom shows the coefficients of the quotient and the remainder.

STEP 4

Check the signs of the coefficients from the synthetic division: $1, 19, 96, 484, 2459$ .
All coefficients are non-negative.

STEP 5

Perform synthetic division of $f(x) = x^{4} + 14x^{3} + x^{2} + 4x + 39$ by $x + 5$ (equivalent to $x - (-5)$ ).
Write down the coefficients of $f(x)$ : $1, 14, 1, 4, 39$ .
Perform synthetic division:
$\begin{array}{r|rrrrr} -5 & 1 & 14 & 1 & 4 & 39 \\ & & -5 & -45 & 220 & -1120 \\ \hline & 1 & 9 & -44 & 224 & -1081 \\ \end{array}$
The row at the bottom shows the coefficients of the quotient and the remainder.

STEP 6

Check the signs of the coefficients from the synthetic division: $1, 9, -44, 224, -1081$ .
The signs alternate: positive, positive, negative, positive, negative.
Conclusion: (a) Since all coefficients from the synthetic division by $x - 5$ are non-negative, 5 is an upper bound for the real zeros of $f(x)$ .
(b) Since the coefficients from the synthetic division by $x + 5$ alternate in sign, -5 is a lower bound for the real zeros of $f(x)$ .

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