Polynomial function: (a) List each real zero and its...

Polynomial function: (a) List each real zero and its multiplicity. (b) Determine whether the graph crosses or touches the $x$ -axis at each $x$ -intercept. (c) Determine the maximum number of turning points on the graph. (d) Determine the end behavior; that is, find the power function that the graph of f resembles for large values of $|x|$. $$ f(x)=7\left(x^{2}+4\right)^{2}(x-5)^{3} $$

Polynomial function:
(a) List each real zero and its multiplicity.
(b) Determine whether the graph crosses or touches the $x$ -axis at each $x$ -intercept.
(c) Determine the maximum number of turning points on the graph.
(d) Determine the end behavior; that is, find the power function that the graph of f resembles for large values of $|x|$.
$$
f(x)=7\left(x^{2}+4\right)^{2}(x-5)^{3}
$$

Key Concepts

Maximum Number of Turning Points

The maximum number of turning points in a polynomial function is determined by its degree, as it can have at most (degree - 1) turning points. This concept is rooted in the fundamental theorem of algebra and the behavior of polynomial functions, providing a bound on the number of local extrema the graph can exhibit.

Real Zeros and Their Multiplicities

This concept involves identifying the values of x that make the polynomial equal to zero and determining the number of times each factor occurs. It requires analyzing the factored form of the polynomial to see which factors yield real roots and what their multiplicities are. Multiplicity affects the graph’s behavior at the corresponding x-intercept, indicating whether the zero is repeated or simple.

Graph Behavior at X-intercepts (Crossing versus Touching)

This concept explains how the multiplicity of a zero affects the way a graph interacts with the x-axis. At an x-intercept, if the corresponding zero has an odd multiplicity, the graph will cross the x-axis; if it has an even multiplicity, it will merely touch the x-axis and turn around. Understanding this behavior helps in sketching the qualitative shape of the graph.

Asymptotic End Behavior of Polynomial Functions

This concept uses the leading term of the polynomial, which dominates the behavior for large absolute values of x, to determine the end behavior of the function. The degree and the sign of the leading coefficient dictate whether the function rises or falls as x approaches positive or negative infinity. Understanding this allows one to predict the general shape and growth of the function's graph at its extreme ends.

Transcript

00:01 Now we are looking at exercise 61, and we have the function f of x equals 7 times the quantity of x squared plus 4 squared times the quantity of x minus 5 cubed.

00:20 So for part a, we want to find the real zero.

00:23 So as you can see on the right, i've explained that we want to set each, the inside of each parentheses, equal to zero and solve.

00:33 Because we're basically using the zero product property.

00:40 And any real zero that we get from these, from solving, will be our answer.

00:51 So for x squared plus 4 equals 0, i would subtract 4 on both sides, and i would square root.

01:07 However, when you square root a negative number, you get non -real solution.

01:16 So i don't even need to continue that anymore.

01:20 Now, if i set x minus 5 equals 0, i can add 5 to both sides, and i get x equals 5.

01:29 So the real zero here, the only real zero that we have here, is x equals 5.

01:40 And to determine the multiplicity, i just need to look at the exponent on that factor.

01:50 So this has multiplicity.

01:57 Now, part b says to determine whether the graph touches, or crosses the x -axis at those 0.

02:09 So since the multiplicity is odd, it is going to cross the x -axis.

02:31 And once again, that's because the 3, right, multiplicity 3 is odd...

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