Algebra Examples

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

Step 1

Identify the degree of the function.

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Step 1.1

Simplify and reorder the polynomial.

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Step 1.1.1

Simplify by multiplying through.

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Step 1.1.1.1

Apply the distributive property.

$(- x^{2} - 1 \cdot 1) (2 x^{4} - 3)$

Step 1.1.1.2

Multiply $- 1$ by $1$ .

$(- x^{2} - 1) (2 x^{4} - 3)$

Step 1.1.2

Expand $(- x^{2} - 1) (2 x^{4} - 3)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

$- x^{2} (2 x^{4} - 3) - 1 (2 x^{4} - 3)$

Step 1.1.2.2

Apply the distributive property.

$- x^{2} (2 x^{4}) - x^{2} \cdot - 3 - 1 (2 x^{4} - 3)$

Step 1.1.2.3

Apply the distributive property.

$- x^{2} (2 x^{4}) - x^{2} \cdot - 3 - 1 (2 x^{4}) - 1 \cdot - 3$

Step 1.1.3

Simplify each term.

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Step 1.1.3.1

Rewrite using the commutative property of multiplication.

$- 1 \cdot 2 x^{2} x^{4} - x^{2} \cdot - 3 - 1 (2 x^{4}) - 1 \cdot - 3$

Step 1.1.3.2

Multiply $x^{2}$ by $x^{4}$ by adding the exponents.

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Step 1.1.3.2.1

Move $x^{4}$ .

$- 1 \cdot 2 (x^{4} x^{2}) - x^{2} \cdot - 3 - 1 (2 x^{4}) - 1 \cdot - 3$

Step 1.1.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 1 \cdot 2 x^{4 + 2} - x^{2} \cdot - 3 - 1 (2 x^{4}) - 1 \cdot - 3$

Step 1.1.3.2.3

Add $4$ and $2$ .

$- 1 \cdot 2 x^{6} - x^{2} \cdot - 3 - 1 (2 x^{4}) - 1 \cdot - 3$

Step 1.1.3.3

Multiply $- 1$ by $2$ .

$- 2 x^{6} - x^{2} \cdot - 3 - 1 (2 x^{4}) - 1 \cdot - 3$

Step 1.1.3.4

Multiply $- 3$ by $- 1$ .

$- 2 x^{6} + 3 x^{2} - 1 (2 x^{4}) - 1 \cdot - 3$

Step 1.1.3.5

Multiply $2$ by $- 1$ .

$- 2 x^{6} + 3 x^{2} - 2 x^{4} - 1 \cdot - 3$

Step 1.1.3.6

Multiply $- 1$ by $- 3$ .

$- 2 x^{6} + 3 x^{2} - 2 x^{4} + 3$

Step 1.2

The largest exponent is the degree of the polynomial.

$6$

Step 2

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

Even

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

Simplify by multiplying through.

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Step 3.1.1.1

Apply the distributive property.

$(- x^{2} - 1 \cdot 1) (2 x^{4} - 3)$

Step 3.1.1.2

Multiply $- 1$ by $1$ .

$(- x^{2} - 1) (2 x^{4} - 3)$

Step 3.1.2

Expand $(- x^{2} - 1) (2 x^{4} - 3)$ using the FOIL Method.

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Step 3.1.2.1

Apply the distributive property.

$- x^{2} (2 x^{4} - 3) - 1 (2 x^{4} - 3)$

Step 3.1.2.2

Apply the distributive property.

$- x^{2} (2 x^{4}) - x^{2} \cdot - 3 - 1 (2 x^{4} - 3)$

Step 3.1.2.3

Apply the distributive property.

$- x^{2} (2 x^{4}) - x^{2} \cdot - 3 - 1 (2 x^{4}) - 1 \cdot - 3$

Step 3.1.3

Simplify terms.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

Rewrite using the commutative property of multiplication.

$- 1 \cdot 2 x^{2} x^{4} - x^{2} \cdot - 3 - 1 (2 x^{4}) - 1 \cdot - 3$

Step 3.1.3.1.2

Multiply $x^{2}$ by $x^{4}$ by adding the exponents.

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Step 3.1.3.1.2.1

Move $x^{4}$ .

$- 1 \cdot 2 (x^{4} x^{2}) - x^{2} \cdot - 3 - 1 (2 x^{4}) - 1 \cdot - 3$

Step 3.1.3.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 1 \cdot 2 x^{4 + 2} - x^{2} \cdot - 3 - 1 (2 x^{4}) - 1 \cdot - 3$

Step 3.1.3.1.2.3

Add $4$ and $2$ .

$- 1 \cdot 2 x^{6} - x^{2} \cdot - 3 - 1 (2 x^{4}) - 1 \cdot - 3$

Step 3.1.3.1.3

Multiply $- 1$ by $2$ .

$- 2 x^{6} - x^{2} \cdot - 3 - 1 (2 x^{4}) - 1 \cdot - 3$

Step 3.1.3.1.4

Multiply $- 3$ by $- 1$ .

$- 2 x^{6} + 3 x^{2} - 1 (2 x^{4}) - 1 \cdot - 3$

Step 3.1.3.1.5

Multiply $2$ by $- 1$ .

$- 2 x^{6} + 3 x^{2} - 2 x^{4} - 1 \cdot - 3$

Step 3.1.3.1.6

Multiply $- 1$ by $- 3$ .

$- 2 x^{6} + 3 x^{2} - 2 x^{4} + 3$

Step 3.1.3.2

Move $3 x^{2}$ .

$- 2 x^{6} - 2 x^{4} + 3 x^{2} + 3$

Step 3.2

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

$- 2 x^{6}$

Step 3.3

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

$- 2$

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.

Falls to the left and falls to the right

Step 7