Section+3.1

The substitute quickly explained that a root of a polynomial P(x)is a number (x) such that P(x)=0. The fundamental theorem of algebra states that a polynomial P(x) of degree //n// has //n// roots, some of which may be degenerate.

For example: has 2 roots (both of which are visible) as shown on the graph below because the greatest degree of the polynomial is 2.



We then did For You to Explore problems in the textbook.

__§3.1 FYE #2__

Find, if possible, a cubic polynomial function with a graph that satisfies these conditions


 * The graph crosses the x-axis at (-5,0), (-1, 0), and somewhere on the positive x-axis.
 * From left to right, the graph rises, falls, and rises.
 * The graph crosses the y-axis at (0, -7).

In general, the variable "a" is a root of the function P(x) if and only if (x-a) is a factor of P(x). Since we know that one of the parameters of the function is that one of the points is (-5, 0), this means that P(-5)=0. When the x-coordinate outputs 0 on the y-coordinate, that is a root.

Another thing we know from the information given above is that the graph rises, falls, and rises. Therefore,



The last piece of information given in the book says that the graph passes through the point (0, -7). So, P(0)= -7. With this information you can plug these values into the function to determine what ka is equal to. Once you know what the product of k and a are, you can plug in two quantities that give you this.



__§3.1 FYE #3__

Find, if possible, a cubic polynomial function with a graph that satisfies these conditions


 * The graph crosses the line with equation y=3 at (-5,3), (-1,3), and somewhere with a positive x-coordinate.
 * From left to right, the graph rises, falls, and rises.
 * The graph crosses the y-axis at (0,-4).

p(-5)=3 p(-1)=3 p(a)=3

f(x)=P(x)-3, which is a cubic polynomial f(-5)=3-3=0 f(-1)=0 f(a)=0

f(0)=P(0)-3= -4-3-7

The function f(x) has all of the properties of the polynomial in #2. Therefore, P(x)=f(x)+3. The polynomial function can then be written like the function in #2 plus three because there is a vertical shift of three.