Question

Consider the following polynomial. g(x)=x³-9x²+14x+24. Step 2 of 3: Use synthetic division to identify integer bounds of the real zeros. Find the least upper bound and the greatest lower bound guaranteed by the upper and lower bounds of zeros theorem.

Answer 1

The real zeros of the polynomial g(x)=x^3-9x^2+14x+24 can be identified within bounds using synthetic division, testing factors of the constant term against the leading coefficient to determine the least upper bound and greatest lower bound.

Step-by-step explanation:

The polynomial given is g(x)=x^3-9x^2+14x+24. To find the bounds of the real zeros, we can use synthetic division and the upper and lower bounds theorem. The theorem suggests that possible rational zeros of the polynomial are the factors of the constant term (in this case, 24) divided by the factors of the leading coefficient (in this case, 1). Thus, the factors of 24 are ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24. We can use synthetic division to test these values to find which are actually zeros of the polynomial.

To find the least upper bound, we test positive factors starting from the smallest. We stop at the smallest positive integer for which the synthetic division yields a non-negative set of remainders. Conversely, to find the greatest lower bound, we test the negative factors starting from the highest absolute value, until the synthetic division provides a non-positive set of remainders.