Find the End Behavior f(x)=1/2x(x+5)(x-3)

$f (x) = \frac{1}{2} x (x + 5) (x - 3)$

Step 1

Identify the degree of the function.

Simplify and reorder the polynomial.

Simplify by multiplying through.

Apply the distributive property.

$\frac{1}{2} \cdot ((x \cdot x + x \cdot 5) (x - 3))$

Simplify the expression.

Multiply $x$ by $x$ .

$\frac{1}{2} \cdot ((x^{2} + x \cdot 5) (x - 3))$

Move $5$ to the left of $x$ .

$\frac{1}{2} \cdot ((x^{2} + 5 \cdot x) (x - 3))$

Expand $(x^{2} + 5 x) (x - 3)$ using the FOIL Method.

Apply the distributive property.

$\frac{1}{2} \cdot (x^{2} (x - 3) + 5 x (x - 3))$

Apply the distributive property.

$\frac{1}{2} \cdot (x^{2} x + x^{2} \cdot - 3 + 5 x (x - 3))$

Apply the distributive property.

$\frac{1}{2} \cdot (x^{2} x + x^{2} \cdot - 3 + 5 x \cdot x + 5 x \cdot - 3)$

Simplify and combine like terms.

Simplify each term.

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

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

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

$\frac{1}{2} \cdot (x^{2} x^{1} + x^{2} \cdot - 3 + 5 x \cdot x + 5 x \cdot - 3)$

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

$\frac{1}{2} \cdot (x^{2 + 1} + x^{2} \cdot - 3 + 5 x \cdot x + 5 x \cdot - 3)$

Add $2$ and $1$ .

$\frac{1}{2} \cdot (x^{3} + x^{2} \cdot - 3 + 5 x \cdot x + 5 x \cdot - 3)$

Move $- 3$ to the left of $x^{2}$ .

$\frac{1}{2} \cdot (x^{3} - 3 \cdot x^{2} + 5 x \cdot x + 5 x \cdot - 3)$

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

Move $x$ .

$\frac{1}{2} \cdot (x^{3} - 3 x^{2} + 5 (x \cdot x) + 5 x \cdot - 3)$

Multiply $x$ by $x$ .

$\frac{1}{2} \cdot (x^{3} - 3 x^{2} + 5 x^{2} + 5 x \cdot - 3)$

Multiply $- 3$ by $5$ .

$\frac{1}{2} \cdot (x^{3} - 3 x^{2} + 5 x^{2} - 15 x)$

Add $- 3 x^{2}$ and $5 x^{2}$ .

$\frac{1}{2} \cdot (x^{3} + 2 x^{2} - 15 x)$

Apply the distributive property.

$\frac{1}{2} x^{3} + \frac{1}{2} (2 x^{2}) + \frac{1}{2} (- 15 x)$

Simplify.

Combine $\frac{1}{2}$ and $x^{3}$ .

$\frac{x^{3}}{2} + \frac{1}{2} (2 x^{2}) + \frac{1}{2} (- 15 x)$

Cancel the common factor of $2$ .

Factor $2$ out of $2 x^{2}$ .

$\frac{x^{3}}{2} + \frac{1}{2} (2 (x^{2})) + \frac{1}{2} (- 15 x)$

Cancel the common factor.

$\frac{x^{3}}{2} + \frac{1}{2} (2 x^{2}) + \frac{1}{2} (- 15 x)$

Rewrite the expression.

$\frac{x^{3}}{2} + x^{2} + \frac{1}{2} (- 15 x)$

Multiply $\frac{1}{2} (- 15 x)$ .

Combine $- 15$ and $\frac{1}{2}$ .

$\frac{x^{3}}{2} + x^{2} + \frac{- 15}{2} x$

Combine $\frac{- 15}{2}$ and $x$ .

$\frac{x^{3}}{2} + x^{2} + \frac{- 15 x}{2}$

Move the negative in front of the fraction.

$\frac{x^{3}}{2} + x^{2} - \frac{15 x}{2}$

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.

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

Simplify by multiplying through.

Apply the distributive property.

$\frac{1}{2} \cdot ((x \cdot x + x \cdot 5) (x - 3))$

Simplify the expression.

Multiply $x$ by $x$ .

$\frac{1}{2} \cdot ((x^{2} + x \cdot 5) (x - 3))$

Move $5$ to the left of $x$ .

$\frac{1}{2} \cdot ((x^{2} + 5 \cdot x) (x - 3))$

Expand $(x^{2} + 5 x) (x - 3)$ using the FOIL Method.

Apply the distributive property.

$\frac{1}{2} \cdot (x^{2} (x - 3) + 5 x (x - 3))$

Apply the distributive property.

$\frac{1}{2} \cdot (x^{2} x + x^{2} \cdot - 3 + 5 x (x - 3))$

Apply the distributive property.

$\frac{1}{2} \cdot (x^{2} x + x^{2} \cdot - 3 + 5 x \cdot x + 5 x \cdot - 3)$

Simplify and combine like terms.

Simplify each term.

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

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

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

$\frac{1}{2} \cdot (x^{2} x^{1} + x^{2} \cdot - 3 + 5 x \cdot x + 5 x \cdot - 3)$

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

$\frac{1}{2} \cdot (x^{2 + 1} + x^{2} \cdot - 3 + 5 x \cdot x + 5 x \cdot - 3)$

Add $2$ and $1$ .

$\frac{1}{2} \cdot (x^{3} + x^{2} \cdot - 3 + 5 x \cdot x + 5 x \cdot - 3)$

Move $- 3$ to the left of $x^{2}$ .

$\frac{1}{2} \cdot (x^{3} - 3 \cdot x^{2} + 5 x \cdot x + 5 x \cdot - 3)$

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

Move $x$ .

$\frac{1}{2} \cdot (x^{3} - 3 x^{2} + 5 (x \cdot x) + 5 x \cdot - 3)$

Multiply $x$ by $x$ .

$\frac{1}{2} \cdot (x^{3} - 3 x^{2} + 5 x^{2} + 5 x \cdot - 3)$

Multiply $- 3$ by $5$ .

$\frac{1}{2} \cdot (x^{3} - 3 x^{2} + 5 x^{2} - 15 x)$

Add $- 3 x^{2}$ and $5 x^{2}$ .

$\frac{1}{2} \cdot (x^{3} + 2 x^{2} - 15 x)$

Apply the distributive property.

$\frac{1}{2} x^{3} + \frac{1}{2} (2 x^{2}) + \frac{1}{2} (- 15 x)$

Simplify.

Combine $\frac{1}{2}$ and $x^{3}$ .

$\frac{x^{3}}{2} + \frac{1}{2} (2 x^{2}) + \frac{1}{2} (- 15 x)$

Cancel the common factor of $2$ .

Factor $2$ out of $2 x^{2}$ .

$\frac{x^{3}}{2} + \frac{1}{2} (2 (x^{2})) + \frac{1}{2} (- 15 x)$

Cancel the common factor.

$\frac{x^{3}}{2} + \frac{1}{2} (2 x^{2}) + \frac{1}{2} (- 15 x)$

Rewrite the expression.

$\frac{x^{3}}{2} + x^{2} + \frac{1}{2} (- 15 x)$

Multiply $\frac{1}{2} (- 15 x)$ .

Combine $- 15$ and $\frac{1}{2}$ .

$\frac{x^{3}}{2} + x^{2} + \frac{- 15}{2} x$

Combine $\frac{- 15}{2}$ and $x$ .

$\frac{x^{3}}{2} + x^{2} + \frac{- 15 x}{2}$

Move the negative in front of the fraction.

$\frac{x^{3}}{2} + x^{2} - \frac{15 x}{2}$

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

$\frac{x^{3}}{2}$

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

$\frac{1}{2}$

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

Find the End Behavior f(x)=1/2x(x+5)(x-3)

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