Algebra Examples

$f (x) = 4 x (x - 4) (x + 2)$

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

Apply the distributive property.

$(4 x \cdot x + 4 x \cdot - 4) (x + 2)$

Step 1.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.2.1

Move $x$ .

$(4 (x \cdot x) + 4 x \cdot - 4) (x + 2)$

Step 1.1.2.2

Multiply $x$ by $x$ .

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

Step 1.1.3

Multiply $- 4$ by $4$ .

$(4 x^{2} - 16 x) (x + 2)$

Step 1.1.4

Expand $(4 x^{2} - 16 x) (x + 2)$ using the FOIL Method.

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Step 1.1.4.1

Apply the distributive property.

$4 x^{2} (x + 2) - 16 x (x + 2)$

Step 1.1.4.2

Apply the distributive property.

$4 x^{2} x + 4 x^{2} \cdot 2 - 16 x (x + 2)$

Step 1.1.4.3

Apply the distributive property.

$4 x^{2} x + 4 x^{2} \cdot 2 - 16 x \cdot x - 16 x \cdot 2$

Step 1.1.5

Simplify and combine like terms.

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Step 1.1.5.1

Simplify each term.

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Step 1.1.5.1.1

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

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Step 1.1.5.1.1.1

Move $x$ .

$4 (x \cdot x^{2}) + 4 x^{2} \cdot 2 - 16 x \cdot x - 16 x \cdot 2$

Step 1.1.5.1.1.2

Multiply $x$ by $x^{2}$ .

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Step 1.1.5.1.1.2.1

Raise $x$ to the power of $1$ .

$4 (x^{1} x^{2}) + 4 x^{2} \cdot 2 - 16 x \cdot x - 16 x \cdot 2$

Step 1.1.5.1.1.2.2

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

$4 x^{1 + 2} + 4 x^{2} \cdot 2 - 16 x \cdot x - 16 x \cdot 2$

Step 1.1.5.1.1.3

Add $1$ and $2$ .

$4 x^{3} + 4 x^{2} \cdot 2 - 16 x \cdot x - 16 x \cdot 2$

Step 1.1.5.1.2

Multiply $2$ by $4$ .

$4 x^{3} + 8 x^{2} - 16 x \cdot x - 16 x \cdot 2$

Step 1.1.5.1.3

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.5.1.3.1

Move $x$ .

$4 x^{3} + 8 x^{2} - 16 (x \cdot x) - 16 x \cdot 2$

Step 1.1.5.1.3.2

Multiply $x$ by $x$ .

$4 x^{3} + 8 x^{2} - 16 x^{2} - 16 x \cdot 2$

Step 1.1.5.1.4

Multiply $2$ by $- 16$ .

$4 x^{3} + 8 x^{2} - 16 x^{2} - 32 x$

Step 1.1.5.2

Subtract $16 x^{2}$ from $8 x^{2}$ .

$4 x^{3} - 8 x^{2} - 32 x$

Step 1.2

The largest exponent is the degree of the polynomial.

$3$

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

Apply the distributive property.

$(4 x \cdot x + 4 x \cdot - 4) (x + 2)$

Step 3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 3.1.2.1

Move $x$ .

$(4 (x \cdot x) + 4 x \cdot - 4) (x + 2)$

Step 3.1.2.2

Multiply $x$ by $x$ .

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

Step 3.1.3

Multiply $- 4$ by $4$ .

$(4 x^{2} - 16 x) (x + 2)$

Step 3.1.4

Expand $(4 x^{2} - 16 x) (x + 2)$ using the FOIL Method.

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Step 3.1.4.1

Apply the distributive property.

$4 x^{2} (x + 2) - 16 x (x + 2)$

Step 3.1.4.2

Apply the distributive property.

$4 x^{2} x + 4 x^{2} \cdot 2 - 16 x (x + 2)$

Step 3.1.4.3

Apply the distributive property.

$4 x^{2} x + 4 x^{2} \cdot 2 - 16 x \cdot x - 16 x \cdot 2$

Step 3.1.5

Simplify and combine like terms.

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Step 3.1.5.1

Simplify each term.

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Step 3.1.5.1.1

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

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Step 3.1.5.1.1.1

Move $x$ .

$4 (x \cdot x^{2}) + 4 x^{2} \cdot 2 - 16 x \cdot x - 16 x \cdot 2$

Step 3.1.5.1.1.2

Multiply $x$ by $x^{2}$ .

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Step 3.1.5.1.1.2.1

Raise $x$ to the power of $1$ .

$4 (x^{1} x^{2}) + 4 x^{2} \cdot 2 - 16 x \cdot x - 16 x \cdot 2$

Step 3.1.5.1.1.2.2

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

$4 x^{1 + 2} + 4 x^{2} \cdot 2 - 16 x \cdot x - 16 x \cdot 2$

Step 3.1.5.1.1.3

Add $1$ and $2$ .

$4 x^{3} + 4 x^{2} \cdot 2 - 16 x \cdot x - 16 x \cdot 2$

Step 3.1.5.1.2

Multiply $2$ by $4$ .

$4 x^{3} + 8 x^{2} - 16 x \cdot x - 16 x \cdot 2$

Step 3.1.5.1.3

Multiply $x$ by $x$ by adding the exponents.

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Step 3.1.5.1.3.1

Move $x$ .

$4 x^{3} + 8 x^{2} - 16 (x \cdot x) - 16 x \cdot 2$

Step 3.1.5.1.3.2

Multiply $x$ by $x$ .

$4 x^{3} + 8 x^{2} - 16 x^{2} - 16 x \cdot 2$

Step 3.1.5.1.4

Multiply $2$ by $- 16$ .

$4 x^{3} + 8 x^{2} - 16 x^{2} - 32 x$

Step 3.1.5.2

Subtract $16 x^{2}$ from $8 x^{2}$ .

$4 x^{3} - 8 x^{2} - 32 x$

Step 3.2

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

$4 x^{3}$

Step 3.3

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

$4$

Step 4

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

Positive

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 rises to the right

Step 7