Find the Behavior (Leading Coefficient Test) f(x)=-5x(2x-5)^2

$f (x) = - 5 x {(2 x - 5)}^{2}$

Identify the degree of the function.

Simplify and reorder the polynomial.

Rewrite ${(2 x - 5)}^{2}$ as $(2 x - 5) (2 x - 5)$ .

$- 5 x ((2 x - 5) (2 x - 5))$

Expand $(2 x - 5) (2 x - 5)$ using the FOIL Method.

Apply the distributive property.

$- 5 x (2 x (2 x - 5) - 5 (2 x - 5))$

Apply the distributive property.

$- 5 x (2 x (2 x) + 2 x \cdot - 5 - 5 (2 x - 5))$

Apply the distributive property.

$- 5 x (2 x (2 x) + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5)$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$- 5 x (2 \cdot 2 x \cdot x + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5)$

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

Move $x$ .

$- 5 x (2 \cdot 2 (x \cdot x) + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5)$

Multiply $x$ by $x$ .

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

Multiply $2$ by $2$ .

$- 5 x (4 x^{2} + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5)$

Multiply $- 5$ by $2$ .

$- 5 x (4 x^{2} - 10 x - 5 (2 x) - 5 \cdot - 5)$

Multiply $2$ by $- 5$ .

$- 5 x (4 x^{2} - 10 x - 10 x - 5 \cdot - 5)$

Multiply $- 5$ by $- 5$ .

$- 5 x (4 x^{2} - 10 x - 10 x + 25)$

Subtract $10 x$ from $- 10 x$ .

$- 5 x (4 x^{2} - 20 x + 25)$

Apply the distributive property.

$- 5 x (4 x^{2}) - 5 x (- 20 x) - 5 x \cdot 25$

Simplify.

Rewrite using the commutative property of multiplication.

$- 5 \cdot 4 x \cdot x^{2} - 5 x (- 20 x) - 5 x \cdot 25$

Rewrite using the commutative property of multiplication.

$- 5 \cdot 4 x \cdot x^{2} - 5 \cdot - 20 x \cdot x - 5 x \cdot 25$

Multiply $25$ by $- 5$ .

$- 5 \cdot 4 x \cdot x^{2} - 5 \cdot - 20 x \cdot x - 125 x$

Simplify each term.

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

Move $x^{2}$ .

$- 5 \cdot 4 (x^{2} x) - 5 \cdot - 20 x \cdot x - 125 x$

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

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

$- 5 \cdot 4 (x^{2} x^{1}) - 5 \cdot - 20 x \cdot x - 125 x$

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

$- 5 \cdot 4 x^{2 + 1} - 5 \cdot - 20 x \cdot x - 125 x$

Add $2$ and $1$ .

$- 5 \cdot 4 x^{3} - 5 \cdot - 20 x \cdot x - 125 x$

Multiply $- 5$ by $4$ .

$- 20 x^{3} - 5 \cdot - 20 x \cdot x - 125 x$

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

Move $x$ .

$- 20 x^{3} - 5 \cdot - 20 (x \cdot x) - 125 x$

Multiply $x$ by $x$ .

$- 20 x^{3} - 5 \cdot - 20 x^{2} - 125 x$

Multiply $- 5$ by $- 20$ .

$- 20 x^{3} + 100 x^{2} - 125 x$

The largest exponent is the degree of the polynomial.

$3$

Since the degree is odd, the ends of the function will point in the opposite directions.

Odd

Identify the leading coefficient.

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

Rewrite ${(2 x - 5)}^{2}$ as $(2 x - 5) (2 x - 5)$ .

$- 5 x ((2 x - 5) (2 x - 5))$

Expand $(2 x - 5) (2 x - 5)$ using the FOIL Method.

Apply the distributive property.

$- 5 x (2 x (2 x - 5) - 5 (2 x - 5))$

Apply the distributive property.

$- 5 x (2 x (2 x) + 2 x \cdot - 5 - 5 (2 x - 5))$

Apply the distributive property.

$- 5 x (2 x (2 x) + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5)$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$- 5 x (2 \cdot 2 x \cdot x + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5)$

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

Move $x$ .

$- 5 x (2 \cdot 2 (x \cdot x) + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5)$

Multiply $x$ by $x$ .

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

Multiply $2$ by $2$ .

$- 5 x (4 x^{2} + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5)$

Multiply $- 5$ by $2$ .

$- 5 x (4 x^{2} - 10 x - 5 (2 x) - 5 \cdot - 5)$

Multiply $2$ by $- 5$ .

$- 5 x (4 x^{2} - 10 x - 10 x - 5 \cdot - 5)$

Multiply $- 5$ by $- 5$ .

$- 5 x (4 x^{2} - 10 x - 10 x + 25)$

Subtract $10 x$ from $- 10 x$ .

$- 5 x (4 x^{2} - 20 x + 25)$

Apply the distributive property.

$- 5 x (4 x^{2}) - 5 x (- 20 x) - 5 x \cdot 25$

Simplify.

Rewrite using the commutative property of multiplication.

$- 5 \cdot 4 x \cdot x^{2} - 5 x (- 20 x) - 5 x \cdot 25$

Rewrite using the commutative property of multiplication.

$- 5 \cdot 4 x \cdot x^{2} - 5 \cdot - 20 x \cdot x - 5 x \cdot 25$

Multiply $25$ by $- 5$ .

$- 5 \cdot 4 x \cdot x^{2} - 5 \cdot - 20 x \cdot x - 125 x$

Simplify each term.

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

Move $x^{2}$ .

$- 5 \cdot 4 (x^{2} x) - 5 \cdot - 20 x \cdot x - 125 x$

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

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

$- 5 \cdot 4 (x^{2} x^{1}) - 5 \cdot - 20 x \cdot x - 125 x$

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

$- 5 \cdot 4 x^{2 + 1} - 5 \cdot - 20 x \cdot x - 125 x$

Add $2$ and $1$ .

$- 5 \cdot 4 x^{3} - 5 \cdot - 20 x \cdot x - 125 x$

Multiply $- 5$ by $4$ .

$- 20 x^{3} - 5 \cdot - 20 x \cdot x - 125 x$

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

Move $x$ .

$- 20 x^{3} - 5 \cdot - 20 (x \cdot x) - 125 x$

Multiply $x$ by $x$ .

$- 20 x^{3} - 5 \cdot - 20 x^{2} - 125 x$

Multiply $- 5$ by $- 20$ .

$- 20 x^{3} + 100 x^{2} - 125 x$

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

$- 20 x^{3}$

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

$- 20$

Since the leading coefficient is negative, the graph falls to the right.

Negative

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

Determine the behavior.

Rises to the left and falls to the right

Find the Behavior (Leading Coefficient Test) f(x)=-5x(2x-5)^2

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