Graphing Polynomials Factor the polynomial and use the factored form to find the zeros. Then ske

Graphing Polynomials Factor the polynomial and use the...

Key Concepts

End Behavior of Polynomials

The end behavior of a polynomial describes how the function behaves as the variable approaches positive or negative infinity. This behavior is largely determined by the degree of the polynomial and its leading coefficient. Understanding end behavior is crucial when sketching the graph, as it helps predict the shape of the graph at the far left and right ends.

Graphing Polynomials Using Factored Form

Graphing a polynomial using its factored form involves plotting its zeros on the x-axis and analyzing the behavior near each zero based on their multiplicities. This method helps sketch a more accurate picture of the overall graph by considering how the function behaves around the zeros and using the leading coefficient and degree to determine the end behavior of the polynomial.

Polynomial Factoring

Polynomial factoring is the process of rewriting a polynomial as the product of its simpler polynomial factors. This technique simplifies solving equations by breaking a complex expression into manageable pieces and is often achieved using methods such as the factor theorem, synthetic division, or grouping. Factoring enables the identification of the roots and the analysis of the polynomial’s behavior.

Zeros of a Polynomial

The zeros of a polynomial are the values of the variable for which the polynomial equals zero. Finding these zeros involves setting each factor equal to zero and solving, which is a fundamental step in understanding the structure and graph of the polynomial. The zeros help determine where the graph intersects or touches the x-axis.

Multiplicity of Zeros

Multiplicity refers to the number of times a particular zero appears as a root of the polynomial. When a factor is raised to a higher power, it indicates that the corresponding zero has a higher multiplicity, which affects the behavior of the graph near that zero. For example, zeros with even multiplicity typically cause the graph to touch the x-axis and turn around, whereas zeros with odd multiplicity result in the graph crossing the axis.

Transcript

00:02 So if we wanted to factor this this polynomial to find the zeros first, what we have to do is say, okay, well, we have 1 over 8, and we can factor the inside by using, by factoring by grouping.

00:17 And so if we have, let's say, let's remove this 1 over 8 first.

00:21 And so let's look at the first two terms inside the parentheses.

00:25 So if we factor out in x cubed here, we have x cubed times times 2 x.

00:32 Plus 3 and so if we factor out if we factor out a a let's say 8 here a negative 8 then for the next two terms inside the parentheses what we're left with is 2x plus 3 again and so since these two terms are equal well we can rewrite this as as x cubed minus 8 times 2x plus 3 and so this whole thing is is remember we're squaring this so this is squared.

01:08 And so now what we have is 1 over 8 times x cubed minus 8 squared.

01:18 So let's say this is, we can just remove this bracket here.

01:22 This becomes x cubed minus 8 times 2x plus 3.

01:29 And so both of these are squared.

01:31 And so notice here that we have this x will equal to 2.

01:35 X is equal to 2 and this has a multiplicity of 2 and also here we have x is equal to negative 3 over 2 with a multiplicity of 2 and so now let's determine our end behavior here well if x approaches positive infinity if x approaches positive infinity then we have we have that our polynomial our polynomial p of x will also approach positive infinity and so now as x approaches has x as negative infinity are polynomial, again, we'll also approach positive infinity.

02:13 And so because we have these even multiplicities for both of our x intercepts, so then what we can do now is say that we have, let's say, we can graph this x and y.

02:33 And so because of the even multiplicities, we will only touch the x -axis...

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