Use the Rational Zero Theorem to find all real zeros

Question

2x⁴ − 5x³ − 6x² + 20x − 8 = 0

Answer 1

2x⁴ – 5x³ – 6x² + 20x – 8 = (2x⁴ – 6x² – 8) + (–5x³ + 20x)

This grouping is motivated by seeing that both groups are readily factored.

(2x⁴ – 6x² – 8) = 2(x⁴ – 3x² – 4) = 2(x² – 4)(x² + 1) and (–5x³ + 20x) = –5x(x² – 4)

So now we have

2x⁴ – 5x³ – 6x² + 20x – 8 = (2(x² + 1) – 5x)(x² – 4) = (2x² – 5x + 2)(x² – 4) = (2x² – 5x + 2)(x – 2)(x + 2)

2x² – 5x + 2 = 2x² – 4x – x + 2 = 2x(x – 2) – 1(x – 2) = (2x – 1)(x – 2)

Giving us the final factored form

2x⁴ – 5x³ – 6x² + 20x – 8 = (2x – 1)(x – 2)(x – 2)(x + 2) = (2x – 1)(x – 2)²(x + 2)

The real zeros come from solving 2x – 1 = 0, x – 2 = 0, and x + 2 = 0

Answer 2

Use the rational zero (root) theorem to find all the real zeros of:

2x⁴ − 5x³ − 6x² + 20x − 8 = 0

The rational zeros (or roots) must be of the form ±p/q, where p is an integer factor of the constant term and q is an integer factor of the leading coefficient.

So, p = -8 and q = 2.

The set of all possible factors is ± { 1/2, 1, 2, 4, 8 }.

Use either synthetic or polynomial division to determine which of these is an actual factor. You can also compute f(x) for each of these values; if f(x)=0, then that x is a zero.

Because the degree of the polynomial is 4, there are at most 4 zeros. After you have found 4 roots, you are done.

A graphing calculator can also be used to find the roots.

The roots are { -2, 1/2, 2 }. 2 has a multiplicity of 2; in other words, it is a double real root.