What would be the largest power of x if we were to multiply the factors? Remember that when a zero of a polynomial has even multiplicity, the graph touches the x-axis at this point.
Degree: 4
Graph:
Practice makes perfect
We want to state the degree and sketch the graph of the given function.
f(x) = (x+4)(x+1)^2(x-2)
Let's start by finding the degree.
Degree
There are 3 unique factors being multiplied to create the polynomial. Let's take a look at the biggest power of x in each factor.
f(x) = (x+4)^1(x+1)^2(x-2)^1
If we were to multiply the highest powers of x in each factor, we would be multiplying the degrees 1, 2, and 1. Remember that when we multiply powers with the same base, we add exponents.
f(x) = (x+4)^1(x+1)^2(x-2)^1 ⇓
Degree: 1 + 2 + 1 = 4
This means that the degree of f(x) is 4. Now let's graph f(x). To do so, we will find x- and y-intercepts and determine the function's end behavior.
Intercepts
The x-intercepts are the roots of the function. They are the solutions of f(x) = 0. Let's use the Zero Product Property to find the roots of f(x).
(x+4)(x+1)^2(x-2) = 0 ⇕
x+4 = 0 or x+1 = 0 or x-2 = 0
This gives us three roots: x = -4, x = -1, and x = 2. Let's find the multiplicity of each root. This will help us graph the function. The multiplicity of a zero of a function is the number of times the root can be factored from the polynomial.
Root
Multiplicity
-4
1
- 1
2
2
1
In our case, (x+4) and (x-2) appear in the factorization one time each, but (x+1) appears twice because it has a degree of 2. This is why the multiplicities of -4 and 2 are 1 and the multiplicity of -1 is 2. Let's continue by finding the y-intercept. The y-intercept is the value of y when the function crosses the y-axis. Let's find the value of f(x) when x=0.
Determining the End Behavior and Sketching the Function
Let's determine the end behavior of f(x). We know that the leading coefficient of the function is 1 because the coefficient of x in each factor is 1. If we were to multiply these coefficients, the product is still 1. The degree of f(x) is 4.
f(x) = 1(x+4)^1(x+1)^2(x-2)^1
Degree: 4
Since the degree is even and the leading coefficient is positive, the end behavior of the graph is up and up. Let's sketch the intercepts and the end behavior in the coordinate plane.
We are ready to sketch the graph! Remember that if a root has even multiplicity, the graph touches but does not cross the x-axis at that point. If a root has odd multiplicity, the graph crosses the axis at that point.
