Match the polynomial function with one of the graphs I-VI. Give reasons for your choice. T(x)=x^

Match the polynomial function with one of the graphs I-VI....

Key Concepts

Turning Points

The maximum number of turning points in a polynomial function is one less than its degree. Recognizing the number and approximate location of these turning points helps in understanding the overall shape of the graph and matching it with the corresponding graphical representation.

Polynomial Factoring

Factoring is an essential tool for breaking down a polynomial into simpler components. Through factoring, one can identify the zeros and their multiplicities, which helps in predicting and sketching the graph of the polynomial accurately.

Degree and End Behavior

The degree and leading coefficient of a polynomial determine its end behavior. For instance, a polynomial with an even degree and a positive leading coefficient will have both ends of its graph rising toward infinity, meaning as x approaches ±?, the function's value increases without bound.

Zeros and Their Multiplicity

Identifying the zeros of a polynomial, along with their multiplicity, is crucial for understanding where and how the graph intersects the x-axis. Zeros with odd multiplicity typically indicate that the graph crosses the x-axis, while zeros with even multiplicity indicate that the graph merely touches the axis and turns around.

Transcript

00:02 All right, we want to discuss the graph of t of x, which is equal to x to the fourth plus 2x cubed.

00:13 And there's two things that we're going to focus on here to determine the shape of this graph.

00:18 And the first is n behavior, which we will determine by looking at the degree of the polynomial.

00:30 The degree of the polynomial is 4.

00:33 In other words, it's an even degree.

00:36 And it has a positive lead coefficient, which means all of this tells us that as x approaches negative infinity, in other words, as x is going to the left, y is going to approach positive infinity, as x is going to the right towards positive infinity, y is also approaching positive infinity.

01:04 So it tells me that the ends of our graph, whatever it's doing in the middle, the ends of our graph as we go to both the left and the right is going up, they're increasing.

01:15 If we had an odd degree polynomial, then our ends would be going in opposite directions.

01:20 But because it's an even degree polynomial, they're going in the same direction, and in this case, they both happen to be going up because the lead coefficient is positive.

01:29 So next, we want to talk about intercepts, right? our x intercepts, also sometimes referred to as roots and zeros.

01:40 To find those, we have to let t of x equals 0.

01:44 So in other words, we want to know when x to the 4th plus 2x cubed is equal to 0.

01:53 If i can factor out a common factor of x cubed, then i get x plus 2 left over as the other factor, right? i have a greatest common factor here of x cubed.

02:10 I divide out the x cubed.

02:11 X to the 4th divided by x cubed is x.

02:13 2x cubed divided by x cubed is 2.

02:16 At this point, we would let each factor equal zero...

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