Answer to Question #324542 in Algebra for Busi

Question

Use the Upper and Lower Bounds Theorem to show that the real zeros of

(i) P (x) = 2x^3-7x^2+4x+4 lie between −4 and 2.

Answer 1

Upper Bound

If you divide a polynomial function f(x) by (x - c), where c > 0, using synthetic division and this yields all positive numbers, then c is an upper bound to the real roots of the equation f(x) = 0. An upper bound is an integer greater than or equal to the greatest real zero.

Lower Bound

If you divide a polynomial function f(x) by (x - c), where c < 0, using synthetic division and this yields alternating signs, then c is a lower bound to the real roots of the equation f(x) = 0. A lower bound is an integer less than or equal to the least real zero.

Task

Show that the real zeros of P(x) = 2x^3-7x^2+4x+4 lie between −4 and 2.

In other words, we need to show that -4 is a lower bound and 2 is an upper bound for real roots of the given equation.

Checking the Lower Bound:

Lets apply synthetic division with -4 and see if we get alternating signs. (see pic.1)

The values are of the alternating sign. Hence, -4 is the lower boundary.

Checking the Upper Bound:

Lets apply synthetic division with 2 and see if we get all positive. (see pic.2)

We have zero as the last coefficient which means that 2 is the real root of this equation.

The Rational Zeros Theorem says that the possible rational zeros for the function P(x) are factors of 4 divided by factors of 2. These are the numbers ±1, ±2, ±4, ±​1/2. So the only possible rational zero greater than 2 is 4. Lets apply synthetic division with 4 and see what we get. (see pic.3)

All the coefficients and the remainder are non-negative. So 4 is the upper boundary and NOT a real zero of this equation.

Since an upper bound is an integer greater than or equal to the greatest real zero and we already find that the greatest zero is 2 than we can say that 2 is the upper boundary.

Since -4 is a lower bound and 2 is an upper bound for the real roots of the equation, then that means all real roots of the equation lie between -4 and 2.