Ch. 4 Polynomial Writing Project

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Ch. 4 Polynomial Writing Project

Identifying the Number of Real Zeros or Complex Zeros

The number of Real Zeros and Complex Zeros is equivalent to the highest degree of the polynomial

Definitions

End Behavior

The End Behavior of a graph is determined by how the graph is acting on the right side of the origin when it runs off of the graph area. If it is going upwards, then the end behavior is +∞.

Leading Coefficient

The Leading Coefficient is the number in front of the highest ranking exponential variable. When a polynomial is in the correct order in terms of powers, it is the first number.

Turning Points

Turning Points are where the graph swtiches directions, creating minimums and maximums.

X-Intercepts / Zeros

The x-intercepts/zeros are where the graph crosses through the x-axis, and they are the solutions to the graphs equations.

Polynomials

An expression consisting of the sum of two or more terms each of which is the product of a constant and a variable raised to an integral power: ax^2 + bx + c is a polynomial, where a,b, and c are constants and x is a variable.

Solving a Polynomial

If not given any Zeros

If no zeros are given in order to solve for a polynomial, then polynomial must be broken down into as many factorable sections as possible. Then set these equal to zero in order to solve for the zeros.

If given two Real Zeros

Let's say that 3 and 5 are the real zeros that we are given. If 3 is a zero of P(x), then (x - 3) is a factor of P(x). Use polynomial long division or synthetic division to divide P(x) by (x - 3). A factorable quadratic equation will come up. Then, since 5 is a zero of P(x), then (x - 5) is a factor of P(x). Now the new factorable equation is divided by (x - 5) and then that will give another factorable quadratic equation which, when solved, will give you the other zeros of P(x).

If given a Complex Zero

Let's say that (1 + i) is the real zero that we are given. That also means that (1 - i) is also a real zero due. Then setup the factors that are in the polynomial: (x - [1 + i]), (x - [1 - i]), and (x - c). "C" because that is the missing variable that we are solving for. Then foil these factors out to get a polynomial and then set it equal to the original polynomial. to solve for the "C". Then you will need to do either long division or synthetic division in order to solve for "C" for the last step.

If given a Real Zero

Let's say that 3 is the real zero that we are given. If 3 is a zero of P(x), then (x - 3) is a factor of P(x). Use polynomial long division or synthetic division to divide P(x) by (x - 3). The quotient, set equal to zero, will be another factorable quadratic equation which, when solved, will give you the other zeros of P(x).

Identifying Multiplicities

Odd

If a real rool/zero has a multiplicity that is odd, then the graph will bend through the x-axis or "wiggle" on the x-axis. ex.: (x-2)^3. The graph will wiggle the x axis at x= 2.

Even

If a real rool/zero has a multiplicity that is even, then the graph will touch the x-axis or "hug" the x-axis, but not cross it. ex.: (x-2)^2. The graph will hug the x axis at x= 2.

Identifying the Degree of a Polynomial

By End Behavior / Pos. or Neg. Leading Coefficient / Even or Odd Degree

If the end behavior for the -∞ and the +∞ are both pointing in the same direction (either both up or down) the degree is even. However, If the end behavior for the -∞ and the +∞ are both pointing in opposites directions (one up and one down; vice versa) the degree is then odd. If the right side of the graph (to the right side of the origin) is pointing upwards in the +∞ direction, then the leading coefficient is positive and the end behavior is +∞. However, If the right side of the graph is pointing downwards in the -∞ direction, then the leading coefficient is negative and the end behavior is -∞.

By Zeros

The number of zeros in a graphed polynomial is equivalent to the degree of the polynomial.

By Turning Points

The number of turning in a graphed polynomial is equivalent to one less the degree of the polynomial.