#1**+1 **

3) f(x) = -2x^{2}(x - 2)^{3}(x + 4)^{4}(x - 5)

The degree of a polynomial is the number of times that the variable (the 'x') is used as a factor.

in x^{2} = 2 times

in (x - 2)^{3} = 3 times

in (x + 4)^{4} = 4 times

in (x - 5) = 1 time

Total: 2 + 3 + 4 + 1 = 10 times <-- degree = 10

End behavior: What happens at the far left-end of the graph and at the far right-end of the graph.

For left-end behavior, place a "large" negative number (such as -10000) into each factor and see if the factor

becomes positive or negative:

-2 is negative

x^{2} is positive

(x - 2)^{3} is negative

(x + 4)^{4} is positive

(x - 5) is negative

Multiplying together, the answer is negative; so at the far left-end of the graph, the y-value will be negative;

so the graph starts at the lower-left end.

For right-end behavior, place a large positive number (such as 10000) into each factor and see if the factor

is positive or negative:

-2 is negative

x^{2} is positive

(x - 2)^{3} is positive

(x + 4)^{4 } is positive

(x - 5) is positive

Multiplying together, the answer is negative; so at the far right-end of the graph, the y-value will be negative;

so the graph ends at the lower-right end.

Zeros occur at the x-values that make the factors zero.

x^{2} zero at 0

(x - 2)^{3} zero at 2

(x + 4)^{4} zero at -4

(x- 5) zero at 5

Bounces or pass through: the graph bounces at a zero that has an even degree, passes through at a zero that

has an odd degree.

x^{2} bounces at 0

(x - 2)^{3} passes through at 2

(x + 4)^{4} bounces at -4

(x - 5) passes through at 5

4) f(x) = 3x^{7} - 48x^{5}

First, factor this: f(x) = 3x^{7} - 48x^{5} = 3x^{5}(x^{2} - 16) = 3x^{5}(x + 4)(x - 4)

Now, analyze this as problem 4) was analyzed.

geno3141 Apr 20, 2020