71-74= Finding a Polynomial from a Graph Find the polynomial of the specified degree whose graph

71-74= Finding a Polynomial from a Graph Find the polynomial...

Key Concepts

Polynomial Functions

Polynomial functions are algebraic expressions consisting of variables and coefficients combined using only addition, subtraction, multiplication, and non-negative integer exponents. They form the basis for understanding more complex functions and are widely used in modeling various real-world scenarios.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable present in the expression. It determines many important characteristics, including the number of possible turning points of the graph and the overall end behavior as the variable approaches infinity or negative infinity.

Zeros and Their Multiplicities

Zeros of a polynomial are the solutions to the equation where the polynomial equals zero; they represent the x-intercepts of the graph. The multiplicity of a zero indicates how many times a factor is repeated, affecting whether the graph crosses the x-axis or merely touches it at that point.

Leading Coefficient and End Behavior

The leading coefficient is the coefficient of the term with the highest degree and plays a crucial role in determining the end behavior of the polynomial. This leads to predictions about how the polynomial behaves as the input values become very large or very small.

Graphical Analysis of Polynomials

Graphical analysis involves studying the key features of a polynomial's graph, such as intercepts, turning points, and overall shape, to infer the algebraic structure of the polynomial. This process helps in reconstructing the polynomial from its visual representation.

Transcript

00:01 Okay, this is definitely not the prettiest graph in the world, but what we need to focus on is that it intersects at negative 2, negative 1, and then it intersects at 1 with this kind of weird hitting the graph and bouncing back up.

00:15 And so that basically means that we could write x plus 2 as a factor because if i plug in negative 2, it becomes 0.

00:23 It's similar way x plus 1 is also a factor.

00:26 And then x minus 1 is a factor, but in a weird 1.

00:30 Way from a previous chapter we know that if it hits right there and bounces back it has a multiplicity of two or in other words you can square this whole thing so let's rewrite this is x plus and you know what let's go one step right now let's say x times x is x squared i'm just going to simplify these two x squared not exactly the best squared i'm written x squared uh two times x is 2x.

00:57 1 times x is 1x.

00:58 Together that's 3x.

01:00 X squared plus 3x, 2 times 1 is 2.

01:03 2.

01:05 All right, and x minus 1, x minus 1, you know what? let's just distribute that right now over here.

01:12 X minus 1, x minus 1.

01:15 X times x is x squared.

01:19 Negative 1 times x is negative 1x.

01:21 Negative 1 times x is another negative 1x.

01:24 Even that's with negative 2x.

01:26 Negative 1 times negative 1 is a positive 1.

01:30 Okay, let's be careful here as we do this.