Find the End Behavior f(x)=-1/2x(x+5)(x-9)(x+2)

$f (x) = - \frac{1}{2} x (x + 5) (x - 9) (x + 2)$

Step 1

Identify the degree of the function.

Simplify and reorder the polynomial.

Simplify by multiplying through.

Apply the distributive property.

$- \frac{1}{2} \cdot ((x \cdot x + x \cdot 5) (x - 9) (x + 2))$

Simplify the expression.

Multiply $x$ by $x$ .

$- \frac{1}{2} \cdot ((x^{2} + x \cdot 5) (x - 9) (x + 2))$

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

$- \frac{1}{2} \cdot ((x^{2} + 5 \cdot x) (x - 9) (x + 2))$

Expand $(x^{2} + 5 x) (x - 9)$ using the FOIL Method.

Apply the distributive property.

$- \frac{1}{2} \cdot ((x^{2} (x - 9) + 5 x (x - 9)) (x + 2))$

Apply the distributive property.

$- \frac{1}{2} \cdot ((x^{2} x + x^{2} \cdot - 9 + 5 x (x - 9)) (x + 2))$

Apply the distributive property.

$- \frac{1}{2} \cdot ((x^{2} x + x^{2} \cdot - 9 + 5 x \cdot x + 5 x \cdot - 9) (x + 2))$

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 $1$ .

$- \frac{1}{2} \cdot ((x^{2} x^{1} + x^{2} \cdot - 9 + 5 x \cdot x + 5 x \cdot - 9) (x + 2))$

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

$- \frac{1}{2} \cdot ((x^{2 + 1} + x^{2} \cdot - 9 + 5 x \cdot x + 5 x \cdot - 9) (x + 2))$

Add $2$ and $1$ .

$- \frac{1}{2} \cdot ((x^{3} + x^{2} \cdot - 9 + 5 x \cdot x + 5 x \cdot - 9) (x + 2))$

Move $- 9$ to the left of $x^{2}$ .

$- \frac{1}{2} \cdot ((x^{3} - 9 \cdot x^{2} + 5 x \cdot x + 5 x \cdot - 9) (x + 2))$

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

Move $x$ .

$- \frac{1}{2} \cdot ((x^{3} - 9 x^{2} + 5 (x \cdot x) + 5 x \cdot - 9) (x + 2))$

Multiply $x$ by $x$ .

$- \frac{1}{2} \cdot ((x^{3} - 9 x^{2} + 5 x^{2} + 5 x \cdot - 9) (x + 2))$

Multiply $- 9$ by $5$ .

$- \frac{1}{2} \cdot ((x^{3} - 9 x^{2} + 5 x^{2} - 45 x) (x + 2))$

Add $- 9 x^{2}$ and $5 x^{2}$ .

$- \frac{1}{2} \cdot ((x^{3} - 4 x^{2} - 45 x) (x + 2))$

Expand $(x^{3} - 4 x^{2} - 45 x) (x + 2)$ by multiplying each term in the first expression by each term in the second expression.

$- \frac{1}{2} \cdot (x^{3} x + x^{3} \cdot 2 - 4 x^{2} x - 4 x^{2} \cdot 2 - 45 x \cdot x - 45 x \cdot 2)$

Simplify terms.

Simplify each term.

Multiply $x^{3}$ by $x$ by adding the exponents.

Multiply $x^{3}$ by $x$ .

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

$- \frac{1}{2} \cdot (x^{3} x^{1} + x^{3} \cdot 2 - 4 x^{2} x - 4 x^{2} \cdot 2 - 45 x \cdot x - 45 x \cdot 2)$

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

$- \frac{1}{2} \cdot (x^{3 + 1} + x^{3} \cdot 2 - 4 x^{2} x - 4 x^{2} \cdot 2 - 45 x \cdot x - 45 x \cdot 2)$

Add $3$ and $1$ .

$- \frac{1}{2} \cdot (x^{4} + x^{3} \cdot 2 - 4 x^{2} x - 4 x^{2} \cdot 2 - 45 x \cdot x - 45 x \cdot 2)$

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

$- \frac{1}{2} \cdot (x^{4} + 2 \cdot x^{3} - 4 x^{2} x - 4 x^{2} \cdot 2 - 45 x \cdot x - 45 x \cdot 2)$

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

Move $x$ .

$- \frac{1}{2} \cdot (x^{4} + 2 x^{3} - 4 (x \cdot x^{2}) - 4 x^{2} \cdot 2 - 45 x \cdot x - 45 x \cdot 2)$

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

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

$- \frac{1}{2} \cdot (x^{4} + 2 x^{3} - 4 (x^{1} x^{2}) - 4 x^{2} \cdot 2 - 45 x \cdot x - 45 x \cdot 2)$

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

$- \frac{1}{2} \cdot (x^{4} + 2 x^{3} - 4 x^{1 + 2} - 4 x^{2} \cdot 2 - 45 x \cdot x - 45 x \cdot 2)$

Add $1$ and $2$ .

$- \frac{1}{2} \cdot (x^{4} + 2 x^{3} - 4 x^{3} - 4 x^{2} \cdot 2 - 45 x \cdot x - 45 x \cdot 2)$

Multiply $2$ by $- 4$ .

$- \frac{1}{2} \cdot (x^{4} + 2 x^{3} - 4 x^{3} - 8 x^{2} - 45 x \cdot x - 45 x \cdot 2)$

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

Move $x$ .

$- \frac{1}{2} \cdot (x^{4} + 2 x^{3} - 4 x^{3} - 8 x^{2} - 45 (x \cdot x) - 45 x \cdot 2)$

Multiply $x$ by $x$ .

$- \frac{1}{2} \cdot (x^{4} + 2 x^{3} - 4 x^{3} - 8 x^{2} - 45 x^{2} - 45 x \cdot 2)$

Multiply $2$ by $- 45$ .

$- \frac{1}{2} \cdot (x^{4} + 2 x^{3} - 4 x^{3} - 8 x^{2} - 45 x^{2} - 90 x)$

Simplify terms.

Subtract $4 x^{3}$ from $2 x^{3}$ .

$- \frac{1}{2} \cdot (x^{4} - 2 x^{3} - 8 x^{2} - 45 x^{2} - 90 x)$

Subtract $45 x^{2}$ from $- 8 x^{2}$ .

$- \frac{1}{2} \cdot (x^{4} - 2 x^{3} - 53 x^{2} - 90 x)$

Apply the distributive property.

$- \frac{1}{2} x^{4} - \frac{1}{2} (- 2 x^{3}) - \frac{1}{2} (- 53 x^{2}) - \frac{1}{2} (- 90 x)$

Simplify.

Combine $x^{4}$ and $\frac{1}{2}$ .

$- \frac{x^{4}}{2} - \frac{1}{2} (- 2 x^{3}) - \frac{1}{2} (- 53 x^{2}) - \frac{1}{2} (- 90 x)$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$- \frac{x^{4}}{2} + \frac{- 1}{2} (- 2 x^{3}) - \frac{1}{2} (- 53 x^{2}) - \frac{1}{2} (- 90 x)$

Factor $2$ out of $- 2 x^{3}$ .

$- \frac{x^{4}}{2} + \frac{- 1}{2} (2 (- x^{3})) - \frac{1}{2} (- 53 x^{2}) - \frac{1}{2} (- 90 x)$

Cancel the common factor.

$- \frac{x^{4}}{2} + \frac{- 1}{2} (2 (- x^{3})) - \frac{1}{2} (- 53 x^{2}) - \frac{1}{2} (- 90 x)$

Rewrite the expression.

$- \frac{x^{4}}{2} - 1 (- x^{3}) - \frac{1}{2} (- 53 x^{2}) - \frac{1}{2} (- 90 x)$

Multiply $- 1$ by $- 1$ .

$- \frac{x^{4}}{2} + 1 x^{3} - \frac{1}{2} (- 53 x^{2}) - \frac{1}{2} (- 90 x)$

Multiply $x^{3}$ by $1$ .

$- \frac{x^{4}}{2} + x^{3} - \frac{1}{2} (- 53 x^{2}) - \frac{1}{2} (- 90 x)$

Multiply $- \frac{1}{2} (- 53 x^{2})$ .

Multiply $- 53$ by $- 1$ .

$- \frac{x^{4}}{2} + x^{3} + 53 (\frac{1}{2}) x^{2} - \frac{1}{2} (- 90 x)$

Combine $53$ and $\frac{1}{2}$ .

$- \frac{x^{4}}{2} + x^{3} + \frac{53}{2} x^{2} - \frac{1}{2} (- 90 x)$

Combine $\frac{53}{2}$ and $x^{2}$ .

$- \frac{x^{4}}{2} + x^{3} + \frac{53 x^{2}}{2} - \frac{1}{2} (- 90 x)$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$- \frac{x^{4}}{2} + x^{3} + \frac{53 x^{2}}{2} + \frac{- 1}{2} (- 90 x)$

Factor $2$ out of $- 90 x$ .

$- \frac{x^{4}}{2} + x^{3} + \frac{53 x^{2}}{2} + \frac{- 1}{2} (2 (- 45 x))$

Cancel the common factor.

$- \frac{x^{4}}{2} + x^{3} + \frac{53 x^{2}}{2} + \frac{- 1}{2} (2 (- 45 x))$

Rewrite the expression.

$- \frac{x^{4}}{2} + x^{3} + \frac{53 x^{2}}{2} - 1 (- 45 x)$

Multiply $- 45$ by $- 1$ .

$- \frac{x^{4}}{2} + x^{3} + \frac{53 x^{2}}{2} + 45 x$

The largest exponent is the degree of the polynomial.

$4$

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.

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

Simplify by multiplying through.

Apply the distributive property.

$- \frac{1}{2} \cdot ((x \cdot x + x \cdot 5) (x - 9) (x + 2))$

Simplify the expression.

Multiply $x$ by $x$ .

$- \frac{1}{2} \cdot ((x^{2} + x \cdot 5) (x - 9) (x + 2))$

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

$- \frac{1}{2} \cdot ((x^{2} + 5 \cdot x) (x - 9) (x + 2))$

Expand $(x^{2} + 5 x) (x - 9)$ using the FOIL Method.

Apply the distributive property.

$- \frac{1}{2} \cdot ((x^{2} (x - 9) + 5 x (x - 9)) (x + 2))$

Apply the distributive property.

$- \frac{1}{2} \cdot ((x^{2} x + x^{2} \cdot - 9 + 5 x (x - 9)) (x + 2))$

Apply the distributive property.

$- \frac{1}{2} \cdot ((x^{2} x + x^{2} \cdot - 9 + 5 x \cdot x + 5 x \cdot - 9) (x + 2))$

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 $1$ .

$- \frac{1}{2} \cdot ((x^{2} x^{1} + x^{2} \cdot - 9 + 5 x \cdot x + 5 x \cdot - 9) (x + 2))$

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

$- \frac{1}{2} \cdot ((x^{2 + 1} + x^{2} \cdot - 9 + 5 x \cdot x + 5 x \cdot - 9) (x + 2))$

Add $2$ and $1$ .

$- \frac{1}{2} \cdot ((x^{3} + x^{2} \cdot - 9 + 5 x \cdot x + 5 x \cdot - 9) (x + 2))$

Move $- 9$ to the left of $x^{2}$ .

$- \frac{1}{2} \cdot ((x^{3} - 9 \cdot x^{2} + 5 x \cdot x + 5 x \cdot - 9) (x + 2))$

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

Move $x$ .

$- \frac{1}{2} \cdot ((x^{3} - 9 x^{2} + 5 (x \cdot x) + 5 x \cdot - 9) (x + 2))$

Multiply $x$ by $x$ .

$- \frac{1}{2} \cdot ((x^{3} - 9 x^{2} + 5 x^{2} + 5 x \cdot - 9) (x + 2))$

Multiply $- 9$ by $5$ .

$- \frac{1}{2} \cdot ((x^{3} - 9 x^{2} + 5 x^{2} - 45 x) (x + 2))$

Add $- 9 x^{2}$ and $5 x^{2}$ .

$- \frac{1}{2} \cdot ((x^{3} - 4 x^{2} - 45 x) (x + 2))$

Expand $(x^{3} - 4 x^{2} - 45 x) (x + 2)$ by multiplying each term in the first expression by each term in the second expression.

$- \frac{1}{2} \cdot (x^{3} x + x^{3} \cdot 2 - 4 x^{2} x - 4 x^{2} \cdot 2 - 45 x \cdot x - 45 x \cdot 2)$

Simplify terms.

Simplify each term.

Multiply $x^{3}$ by $x$ by adding the exponents.

Multiply $x^{3}$ by $x$ .

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

$- \frac{1}{2} \cdot (x^{3} x^{1} + x^{3} \cdot 2 - 4 x^{2} x - 4 x^{2} \cdot 2 - 45 x \cdot x - 45 x \cdot 2)$

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

$- \frac{1}{2} \cdot (x^{3 + 1} + x^{3} \cdot 2 - 4 x^{2} x - 4 x^{2} \cdot 2 - 45 x \cdot x - 45 x \cdot 2)$

Add $3$ and $1$ .

$- \frac{1}{2} \cdot (x^{4} + x^{3} \cdot 2 - 4 x^{2} x - 4 x^{2} \cdot 2 - 45 x \cdot x - 45 x \cdot 2)$

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

$- \frac{1}{2} \cdot (x^{4} + 2 \cdot x^{3} - 4 x^{2} x - 4 x^{2} \cdot 2 - 45 x \cdot x - 45 x \cdot 2)$

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

Move $x$ .

$- \frac{1}{2} \cdot (x^{4} + 2 x^{3} - 4 (x \cdot x^{2}) - 4 x^{2} \cdot 2 - 45 x \cdot x - 45 x \cdot 2)$

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

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

$- \frac{1}{2} \cdot (x^{4} + 2 x^{3} - 4 (x^{1} x^{2}) - 4 x^{2} \cdot 2 - 45 x \cdot x - 45 x \cdot 2)$

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

$- \frac{1}{2} \cdot (x^{4} + 2 x^{3} - 4 x^{1 + 2} - 4 x^{2} \cdot 2 - 45 x \cdot x - 45 x \cdot 2)$

Add $1$ and $2$ .

$- \frac{1}{2} \cdot (x^{4} + 2 x^{3} - 4 x^{3} - 4 x^{2} \cdot 2 - 45 x \cdot x - 45 x \cdot 2)$

Multiply $2$ by $- 4$ .

$- \frac{1}{2} \cdot (x^{4} + 2 x^{3} - 4 x^{3} - 8 x^{2} - 45 x \cdot x - 45 x \cdot 2)$

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

Move $x$ .

$- \frac{1}{2} \cdot (x^{4} + 2 x^{3} - 4 x^{3} - 8 x^{2} - 45 (x \cdot x) - 45 x \cdot 2)$

Multiply $x$ by $x$ .

$- \frac{1}{2} \cdot (x^{4} + 2 x^{3} - 4 x^{3} - 8 x^{2} - 45 x^{2} - 45 x \cdot 2)$

Multiply $2$ by $- 45$ .

$- \frac{1}{2} \cdot (x^{4} + 2 x^{3} - 4 x^{3} - 8 x^{2} - 45 x^{2} - 90 x)$

Simplify terms.

Subtract $4 x^{3}$ from $2 x^{3}$ .

$- \frac{1}{2} \cdot (x^{4} - 2 x^{3} - 8 x^{2} - 45 x^{2} - 90 x)$

Subtract $45 x^{2}$ from $- 8 x^{2}$ .

$- \frac{1}{2} \cdot (x^{4} - 2 x^{3} - 53 x^{2} - 90 x)$

Apply the distributive property.

$- \frac{1}{2} x^{4} - \frac{1}{2} (- 2 x^{3}) - \frac{1}{2} (- 53 x^{2}) - \frac{1}{2} (- 90 x)$

Simplify.

Combine $x^{4}$ and $\frac{1}{2}$ .

$- \frac{x^{4}}{2} - \frac{1}{2} (- 2 x^{3}) - \frac{1}{2} (- 53 x^{2}) - \frac{1}{2} (- 90 x)$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$- \frac{x^{4}}{2} + \frac{- 1}{2} (- 2 x^{3}) - \frac{1}{2} (- 53 x^{2}) - \frac{1}{2} (- 90 x)$

Factor $2$ out of $- 2 x^{3}$ .

$- \frac{x^{4}}{2} + \frac{- 1}{2} (2 (- x^{3})) - \frac{1}{2} (- 53 x^{2}) - \frac{1}{2} (- 90 x)$

Cancel the common factor.

$- \frac{x^{4}}{2} + \frac{- 1}{2} (2 (- x^{3})) - \frac{1}{2} (- 53 x^{2}) - \frac{1}{2} (- 90 x)$

Rewrite the expression.

$- \frac{x^{4}}{2} - 1 (- x^{3}) - \frac{1}{2} (- 53 x^{2}) - \frac{1}{2} (- 90 x)$

Multiply $- 1$ by $- 1$ .

$- \frac{x^{4}}{2} + 1 x^{3} - \frac{1}{2} (- 53 x^{2}) - \frac{1}{2} (- 90 x)$

Multiply $x^{3}$ by $1$ .

$- \frac{x^{4}}{2} + x^{3} - \frac{1}{2} (- 53 x^{2}) - \frac{1}{2} (- 90 x)$

Multiply $- \frac{1}{2} (- 53 x^{2})$ .

Multiply $- 53$ by $- 1$ .

$- \frac{x^{4}}{2} + x^{3} + 53 (\frac{1}{2}) x^{2} - \frac{1}{2} (- 90 x)$

Combine $53$ and $\frac{1}{2}$ .

$- \frac{x^{4}}{2} + x^{3} + \frac{53}{2} x^{2} - \frac{1}{2} (- 90 x)$

Combine $\frac{53}{2}$ and $x^{2}$ .

$- \frac{x^{4}}{2} + x^{3} + \frac{53 x^{2}}{2} - \frac{1}{2} (- 90 x)$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$- \frac{x^{4}}{2} + x^{3} + \frac{53 x^{2}}{2} + \frac{- 1}{2} (- 90 x)$

Factor $2$ out of $- 90 x$ .

$- \frac{x^{4}}{2} + x^{3} + \frac{53 x^{2}}{2} + \frac{- 1}{2} (2 (- 45 x))$

Cancel the common factor.

$- \frac{x^{4}}{2} + x^{3} + \frac{53 x^{2}}{2} + \frac{- 1}{2} (2 (- 45 x))$

Rewrite the expression.

$- \frac{x^{4}}{2} + x^{3} + \frac{53 x^{2}}{2} - 1 (- 45 x)$

Multiply $- 45$ by $- 1$ .

$- \frac{x^{4}}{2} + x^{3} + \frac{53 x^{2}}{2} + 45 x$

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

$- \frac{x^{4}}{2}$

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

$- \frac{1}{2}$

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

Find the End Behavior f(x)=-1/2x(x+5)(x-9)(x+2)

Related Questions

Support LCMGCF.com by disabling your adblocker.