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Post date: Nov 27, 2013 7:10:16 PM
Notes
When we see a graph of a polynomial, real roots are x-intercepts of the graph of f(x).
Let's look at an example:
When we see a graph of a polynomial with no x-intercepts of the graph of f(x), we are looking at a complex root
Let's look at an example:
Here is how you determine complex or real roots from a graph
We can see from the graph of a polynomial, whether it has real roots or is irreducible over the real numbers. How can we tell algebraically, whether a quadratic polynomial has real or complex roots? The symbol i enters the picture, exactly when the term under the square root in the quadratic formula is negative. This term
is called the discriminant.
Consider the discriminant
of the quadratic polynomial .
If the discriminant is positive, the polynomial has 2 distinct real roots, > 0If the discriminant is negative, the polynomial has 2 complex roots, which form a complex conjugate pair, < 0If the discriminant is zero, the polynomial has one real root of multiplicity 2, = 0
Let's look at an example:
1. Quadratic Equation: y = x² + 2x + 1
a=1 b=2 c=1
the discriminant in this case would be 0, which tells us that there should be one real solution to this equation.
You can also see this from the graph:
2. Quadratic Equation: y = x² + 4x + 5
a=1 b=4 c=5
the discriminant in this case would be -4, which tells us that there are no real solutions to this quadratic equation, the only solution are imaginary.
You can also see this from the graph:
3. Quadratic Equation: y = x² − 1
a=1 b=0 c=-1
the discriminant in this case would be 4, which tells us that we have two real solutions that are rational.
You can see this from the graph:
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