Algebra Examples

f (x) = - 3072 + 2432 x + 128 x^{4} - 576 x^{3} + 256 x^{2} - 8 x^{5}

$f (x) = - 3072 + 2432 x + 128 x^{4} - 576 x^{3} + 256 x^{2} - 8 x^{5}$

Step 1

Identify the degree of the function.

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Step 1.1

Identify the exponents on the variables in each term, and add them together to find the degree of each term.

- 3072 \to 0

$- 3072 \to 0$

2432 x \to 1

$2432 x \to 1$

128 x^{4} \to 4

$128 x^{4} \to 4$

- 576 x^{3} \to 3

$- 576 x^{3} \to 3$

256 x^{2} \to 2

$256 x^{2} \to 2$

- 8 x^{5} \to 5

$- 8 x^{5} \to 5$

Step 1.2

The largest exponent is the degree of the polynomial.

5

$5$

5

$5$

Step 2

Since the degree is odd, the ends of the function will point in the opposite directions.

Odd

Step 3

Identify the leading coefficient.

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Step 3.1

Simplify the polynomial, then reorder it left to right starting with the highest degree term.

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Step 3.1.1

Move

- 3072

$- 3072$ .

2432 x + 128 x^{4} - 576 x^{3} + 256 x^{2} - 8 x^{5} - 3072

$2432 x + 128 x^{4} - 576 x^{3} + 256 x^{2} - 8 x^{5} - 3072$

Step 3.1.2

Move

2432 x

$2432 x$ .

128 x^{4} - 576 x^{3} + 256 x^{2} - 8 x^{5} + 2432 x - 3072

$128 x^{4} - 576 x^{3} + 256 x^{2} - 8 x^{5} + 2432 x - 3072$

Step 3.1.3

Move

256 x^{2}

$256 x^{2}$ .

128 x^{4} - 576 x^{3} - 8 x^{5} + 256 x^{2} + 2432 x - 3072

$128 x^{4} - 576 x^{3} - 8 x^{5} + 256 x^{2} + 2432 x - 3072$

Step 3.1.4

Move

- 576 x^{3}

$- 576 x^{3}$ .

128 x^{4} - 8 x^{5} - 576 x^{3} + 256 x^{2} + 2432 x - 3072

$128 x^{4} - 8 x^{5} - 576 x^{3} + 256 x^{2} + 2432 x - 3072$

Step 3.1.5

Reorder

128 x^{4}

$128 x^{4}$ and

- 8 x^{5}

$- 8 x^{5}$ .

- 8 x^{5} + 128 x^{4} - 576 x^{3} + 256 x^{2} + 2432 x - 3072

$- 8 x^{5} + 128 x^{4} - 576 x^{3} + 256 x^{2} + 2432 x - 3072$

- 8 x^{5} + 128 x^{4} - 576 x^{3} + 256 x^{2} + 2432 x - 3072

$- 8 x^{5} + 128 x^{4} - 576 x^{3} + 256 x^{2} + 2432 x - 3072$

Step 3.2

The leading term in a polynomial is the term with the highest degree.

- 8 x^{5}

$- 8 x^{5}$

Step 3.3

The leading coefficient in a polynomial is the coefficient of the leading term.

- 8

$- 8$

- 8

$- 8$

Step 4

Since the leading coefficient is negative, the graph falls to the right.

Negative

Step 5

Use the degree of the function, as well as the sign of the leading coefficient to determine the behavior.

1. Even and Positive: Rises to the left and rises to the right.

2. Even and Negative: Falls to the left and falls to the right.

3. Odd and Positive: Falls to the left and rises to the right.

4. Odd and Negative: Rises to the left and falls to the right

Step 6

Determine the behavior.

Rises to the left and falls to the right

Step 7