Polynomials with Real Zeros (4.4)

Polynomials with Real Zeros (4.4)

The Zeros (ROOTS or x-intercepts of the graph) can take many
numerical forms. In this section we will only consider Zeros that
are REAL NUMBERS. (In the next section we will look at the
possibility of IMAGINARY or COMPLEX NUMBERS.
These occur when we have a RADICAND (number under
the radical sign) that is NEGATIVE.
When we are able to factor a POLYNOMIAL into
LINEAR TERMS [i.e. "(x+2)(2x-3)(x-5)"], the ZEROS
are RATIONAL NUMBERS. (Fractions and Integers)
If one of our FACTORS is a QUADRATIC, we can
use the QUADRATIC FORMULA. When using the
Quadratic Formula the answers are frequently
IRRATIONAL (involving Radicals that are NOT
PERFECT SQUARES). The RADICANDS are NOT:
4, 9, 16, 25, 36, 49, 64,...
If the ZEROS are in fact RATIONAL then they are
of the form = p/q with p being a factor of the
CONSTANT TERM and q a factor of the
LEADING COEFFICIENT.

HERE ARE MORE EXAMPLES:
f(x) = x³ - 2 x² - x - 2 = (x+1)(x+1)(x-1)
the ROOTS are -1 and 1
g(x) = 5x³ - 4 x² + 7 x - 8 = (x-1)(5 x² + x +8)
the ZEROS are 1 and "use Quad Formula
to find the other possible ROOTS")
k(x) = x³ - 3 x² - 2x + 6 = (x-3)(x² - 2)
Roots are 3 and Radical 2 and
the Opposite of Radical 2))
q(x) = 2x³ - 3 x² - 23x + 12 = (2x-1)(x+3)(x-4)
the Zeros are 1/2,  -3, and 4

Here are three online videos (CLICK on the EXAMPLE!) that show the Quadratic Formula in action:
Example #1            Example #2                     Example #3