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

step 1 So the first thing we have to do is factor out this polynomial. So if we factor out a negative x

Graphing Polynomials Factor the polynomial and use the...

Key Concepts

Multiplicity of Zeros

Multiplicity refers to the number of times a particular zero occurs based on the exponent of the factor in the factored form. It plays a significant role in determining the behavior of the graph at the intersection points, indicating whether the graph merely touches the axis or crosses it.

Polynomial Factoring

Polynomial factoring is the process of rewriting a polynomial as a product of simpler polynomials. This technique is crucial as it not only simplifies expressions but also makes it easier to solve equations by setting each factor equal to zero.

Zeros of a Polynomial

Zeros, or roots, are the solutions to the equation when the polynomial is set equal to zero. Finding the zeros is essential because they indicate where the graph of the polynomial will intersect the x-axis, providing key points for graphing.

Graphing Polynomial Functions

Graphing polynomial functions involves combining information about the zeros, the multiplicities of those zeros, and the overall shape dictated by the degree and leading coefficient. This process allows for an accurate sketch of the function's behavior across its domain.

End Behavior

End behavior describes the way a polynomial function behaves as the input values become very large or very small, typically determined by the leading term. Understanding the end behavior helps in predicting the direction of the arms of the graph.

Transcript

00:01 So the first thing we have to do is factor out this polynomial.

00:05 So if we factor out a negative x, what we're left with is 2x squared, 2x squared plus x, and then minus 1.

00:18 And so we want to make this easier to factor.

00:22 So we're going to multiply this 2 by 1.

00:24 So this is the ac method.

00:26 So we have negative x, and then we're going to change this to x squared plus x minus 2.

00:33 And so this becomes easier to factor because we know that two multiple two numbers that add up to negative one or two numbers that multiply to negative two by add up to one are positive two and negative one.

00:46 So we have this negative x and then x plus two and then x minus one.

00:53 But remember since we since we multiplied by two, we now have to divide by two for both.

00:57 So we have this divide by two here.

01:00 It's divide by two here.

01:01 And so we're left with this, negative x times x plus 1, and then 2x minus 1.

01:10 So now we have, we have our zeros are 0, negative 1, and then if we set 2x minus 1 to be equal to 0, we have x is equal to 1 half.

01:25 And so now we know our 0s, all of these have a multiplicity of 1.

01:29 And so now we can determine our n behavior.

01:31 So as x tends to positive infinity, well, let's see here, we have negative infinity times positive infinity times positive infinity...

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