Find the Behavior (Leading Coefficient Test) x^8+2x^4-4x^3+x^2-1

$x^{8} + 2 x^{4} - 4 x^{3} + x^{2} - 1$

Step 1

Write $x^{8} + 2 x^{4} - 4 x^{3} + x^{2} - 1$ as a function.

$f (x) = x^{8} + 2 x^{4} - 4 x^{3} + x^{2} - 1$

Step 2

Identify the degree of the function.

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

$x^{8} \to 8$

$2 x^{4} \to 4$

$- 4 x^{3} \to 3$

$x^{2} \to 2$

$- 1 \to 0$

The largest exponent is the degree of the polynomial.

$8$

Step 3

Since the degree is even, the ends of the function will point in the same direction.

Even

Step 4

Identify the leading coefficient.

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

$x^{8}$

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

$1$

Step 5

Since the leading coefficient is positive, the graph rises to the right.

Positive

Step 6

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 7

Determine the behavior.

Rises to the left and rises to the right

Step 8