Answer:
Step-by-step explanation:
Theorm-The Fundamental Theorem of Algebra: If P(x) is a polynomial of degree n ≥ 1, then P(x) = 0 has exactly n roots, including multiple and complex roots.
Let's verify that the Fundamental Theorem of Algebra holds for quadratic polynomials.
A quadratic polynomial is a second degree polynomial. According to the Fundamental Theorem of Algebra, the quadratic set = 0 has exactly two roots.
As we have seen, factoring a quadratic equation will result in one of three possible situations.
graph 1
The quadratic may have 2 distinct real roots. This graph crosses the
x-axis in two locations. These graphs may open upward or downward.
graph 2
It may appear that the quadratic has only one real root. But, it actually has one repeated root. This graph is tangent to the x-axis in one location (touching once).
graph 3
The quadratic may have two non-real complex roots called a conjugate pair. This graph will not cross or touch the x-axis, but it will have two roots.
If there are only two colors (let's say blue and red), here's what can happen:
sock #1 is blue
#2 is either blue or red. If blue, it matches #1 and you have a pair.
if red, go to #3
#3-either blue or red. If blue, matches #1. If red, matches #2.
OR sock #1 is red... then just reverse the colors. Basically, if you have three things that can only be in two groups, then even if two of them are different, the last one has to match one of them.
Answer:
(-1,0) (0,4) (1,0)
Step-by-step explanation:
The turning points are also called the critical points. It is where the graph changes direction.
They occur at (-1,0) (0,4) (1,0)
47 -14i
You can work this out in the straight-forward way, or you can recognize that (6-i) is a common factor. In the latter case, you have ...
... = (6-i)(5 + 3-i)
... = (6 -i)(8 -i)
This product of binomials is found in the usual way. Each term of one factor is multiplied by each term of the other factor and the results summed. Of course, i = √-1, so i² = -1.
... = 6·8 -6i -8i +i²
... = 48 -14i -1
... =
_____
A suitable graphing calculator will work these complex number problems easily.
