Algebra II: Factoring
Roots
The solutions to y = f (x) when y = 0 are called the roots of a function ( f (x) is any function). These are the points at which the graph of an equation crosses the x -axis.
Roots of Quadratic Functions
We have already learned to solve for x in ax 2 + bx + c = 0 by factoring ax 2 + bx + c and using the zero product property. Since the roots of a function are the points at which y = 0 , we can find the roots of y = ax 2 + bx + c = 0 by factoring ax 2 + bx + c = 0 and solving for x . We can also find the roots of y = ax 2 + bx + c = 0 using the quadratic formula, and we can find the number of roots using the discriminant.
If a quadratic function has 2 roots--i.e., if it can be factored into 2 distinct binomials or if b 2 -4ac > 0 --then it crosses the x - axis twice. Either the vertex is below the x -axis and the leading coefficient is positive, or the vertex is above the x -axis and the leading coefficient is negative.
If a quadratic function has 1 root (a "double root")--i.e. if it can be factored as the square of a single binomial or if b 2 - 4ac = 0 --then it crosses the x -axis once. The vertex lies on the x -axis, and the leading coefficient can be positive or negative.
If a quadratic function has no roots--i.e. if it cannot be factored or if b 2 -4ac < 0 --then it does not cross the x -axis. Either the vertex is above the x -axis and the leading coefficient is positive, or the vertex is below the x -axis and the leading coefficient is negative. The quadratic equation is said to have 2 imaginary roots.
Roots of Other Polynomial Functions
We can find the roots of other polynomial functions by setting
y = 0
and factoring. For example,
y = x
3 -27 = (x - 3)(x
2 +3x
2 + 9)
has one root (
x = 3
), because there is one value of
x
for which
x - 3 = 0
and no values of
x
for which
x
2 + 3x + 9 = 0
(the
discriminant is negative).
y = 45x
3 +18x
2 - 5x - 2 = (3x + 1)(3x - 1)(5x + 2)
has three roots (
x = - 
,, - 
).
The number of distinct roots of an equation is equal to the number of distinct x -values for which y = 0 .
The maximum number of distinct roots that a polynomial can have is equal to the degree of the polynomial. For example, a polynomial of degree 2 can have no more than 2 distinct roots, a polynomial of degree 3 can have no more than 3 distinct roots, etc.
