The math teacher and cheerleading coach have teamed up to help the students do better on their math test. The cheer coach, using dance move names for the positioning of their arms, yells out polynomial functions with different degrees.
For each position the coach yells out, write the shape by describing the position of your left and right arm.

a1. Constant Function:
a2. Positive Linear Function:
a3. Negative Linear Function:
a4. Positive Quadratic Function:
a5. Negative Quadratic Function:
a6. Positive Cubic Function:
a7. Negative Cubic Function:
a8. Positive Quartic Function:
a9. Negative Quartic Function:

When it comes time to take the test not only do the students have to describe the shape of the polynomial function, you have to find the number of positive and negative real zeros, including complex. Use the equation below:
[tex]f(x)=x^5-3x^4-5x^3+5x^2-6x+8[/tex]

b. Identify all possible rational zeros.
c. How many possible positive real zeros are there? How many possible negative real zeros? How many possible complex zeros?
d. Graph the polynomial to approximate the zeros. What are the rational zeros? Use synthetic division to verify these are correct.
e. Write the polynomial in factor form.
f. What are the complex zeros?

Step-by-step explanation:

a1. The shape will be a vertical or horizontal line.

a2. The shape will be shaped like a diagonal line increasing as we go right.

a3. The shape will be shaped like a diagonal line decreasing as we go right.

a4. The shape will be shaped like a U facing upwards.

a5.The shape will be shaped like a U facing downwards.

a6. The shape will look like a S shape and it increases as we go right.

a7. The shape will look like a S shape and it decreases as We go right.

a8. The shape look like a W shape and it facing upwards.

a9. The shape look a W shape facing downwards.

We are given function.

[tex]x {}^{5} - 3x {}^{4} - 5x {}^{3} + 5x {}^{2} - 6x + 8[/tex]

b. We can test by the Rational Roots Test,

This means a the possible roots are

plus or minus(1,2,4,8).

c. If we apply Descrates Rule of Signs,

d. Use Desmos to Graph the Function. Some roots are (-2,1,4).

e.

[tex](x {}^{2} + 1) (x - 1)(x - 4)(x + 2)[/tex]

f. The complex zeroes are

i and -i

Polynomial [tex]f(x) = x^{5} -3x^{4} - 5x^{3} + 5x^{2} - 6x + 8[/tex] in factor form: (x-1)(x+2)(x-4)(x-i)(x+i)

A polynomial is an expression consisting of indeterminates and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables.

Shape of the graph for the following polynomial:

Finding zeros of the polynomial given:

[tex]f(x) = x^{5} -3x^{4} - 5x^{3} + 5x^{2} - 6x + 8[/tex]

By factor theorem, if f(t) = 0, t is a zero of the polynomial.

Taking t = 1.

f(1) = 1 - 3 - 5 + 5 - 6 + 8 = 0

(x - 1) is a factor of the polynomial f(x).

Divide f(x) by (x-1) using long division to find the other factors.

f(x)/(x-1) = [tex]x^{4} -2x^{3}-7x^{2} -2x-8[/tex] is also a factor of f(x).

Factorizing it further:

g(x) = [tex]x^{4} -2x^{3}-7x^{2} -2x-8[/tex]

g(-2) = 16 + 16 - 28 + 4 - 8 = 0

(x + 2) is a factor of g(x) and thus f(x).

g(x)/(x+2) = [tex]x^{3} - 4x^{2} +x - 4[/tex] is a factor of f(x).

Factorizing it further:

k(x) = [tex]x^{3} - 4x^{2} +x - 4[/tex]

k(4) = 64 - 64 + 4 - 4 = 0

(x - 4) is a factor of k(x) thus of f(x).

k(x)/(x-4) = [tex]x^{2} +1[/tex]

Factorizing it further:

l(x) = [tex]x^{2} +1[/tex] = (x + i)(x - i)

Zeros of f(x) = 1, -2, 4, ±i

Rational zeros :  1, -2, 4

Positive real zeros: 1, 4

Negative real zeros: -2

Complex zeros: ±i

Polynomial in factor form: (x-1)(x+2)(x-4)(x-i)(x+i).

Learn more about polynomial here

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