Question

question 6 (5 points)
for this problem, use the given bound to determine if it is an upper bound, lower bound or neither. show all work for full credit!
$f(x)=2x^3 + 10x^2 - 12x$
$k=3$

Explanation:

Step1: Recall the Upper Bound Test

For a polynomial \( f(x) \) with real coefficients and leading coefficient positive, we use synthetic division with \( k \). If all the numbers in the last row (excluding the remainder) are non - negative, then \( k \) is an upper bound.
First, write the coefficients of \( f(x)=2x^{3}+10x^{2}-12x + 0\) (we add the constant term 0 since there is no constant term in the given polynomial). The coefficients are \( 2,10, - 12,0\).
We perform synthetic division with \( k = 3\):

Bring down the leading coefficient 2.

Step2: Perform Synthetic Division

Multiply the number we just brought down (2) by \( k=3\): \(2\times3 = 6\). Add this to the next coefficient: \(10 + 6=16\).

Multiply this new number (16) by 3: \(16\times3 = 48\). Add this to the next coefficient: \(-12+48 = 36\).

Multiply this new number (36) by 3: \(36\times3=108\). Add this to the last coefficient: \(0 + 108 = 108\).

The last row of the synthetic division (the coefficients of the quotient and the remainder) is \(2,16,36,108\). All of these numbers are non - negative.

Answer:

Since all the numbers in the last row of the synthetic division (excluding the remainder) are non - negative, \(k = 3\) is an upper bound for the roots of \(f(x)=2x^{3}+10x^{2}-12x\).