A. Determine if the upper bound theorem identifies the given...

a. Determine if the upper bound theorem identifies the given number as an upper bound for the real zeros of $f(x)$.
b. Determine if the lower bound theorem identifies the given number as a lower bound for the real zeros of $f(x)$.
$f(x)=x^{5}+6 x^{4}+5 x^{2}+x-3$
a. 2
b. -5

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Key Concepts

Upper Bound Theorem

The Upper Bound Theorem is a result in polynomial algebra that states if a positive number is used as a divisor in synthetic division and all the resulting coefficients (including the remainder) are nonnegative, then that number is an upper bound for the real zeros of the polynomial. This theorem helps narrow down the interval within which all real zeros lie and is a useful tool when solving polynomial equations.

Lower Bound Theorem

The Lower Bound Theorem is a complementary concept to the Upper Bound Theorem, providing a criterion to determine a lower bound for the real zeros of a polynomial. It states that if a number is used (often a negative number) in synthetic division and the resulting coefficients alternate in sign, then that number serves as a lower bound for the real zeros. This theorem assists in establishing the smallest possible values for the zeros.

Synthetic Division

Synthetic division is an efficient, shortcut method for dividing a polynomial by a linear factor of the form (x - c). It is particularly useful in conjunction with the Upper and Lower Bound Theorems because it allows for a quick check of the signs and values of the coefficients that result from the division, thereby helping to determine if a candidate number is indeed an upper or lower bound for the real zeros of the polynomial.

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