Find the Behavior (Leading Coefficient Test) f(x)=x^2(x-4)^3(x+5)

$f (x) = x^{2} {(x - 4)}^{3} (x + 5)$

Step 1

Identify the degree of the function.

Simplify and reorder the polynomial.

Use the Binomial Theorem.

$x^{2} (x^{3} + 3 x^{2} \cdot - 4 + 3 x {(- 4)}^{2} + {(- 4)}^{3}) (x + 5)$

Simplify terms.

Simplify each term.

Multiply $- 4$ by $3$ .

$x^{2} (x^{3} - 12 x^{2} + 3 x {(- 4)}^{2} + {(- 4)}^{3}) (x + 5)$

Raise $- 4$ to the power of $2$ .

$x^{2} (x^{3} - 12 x^{2} + 3 x \cdot 16 + {(- 4)}^{3}) (x + 5)$

Multiply $16$ by $3$ .

$x^{2} (x^{3} - 12 x^{2} + 48 x + {(- 4)}^{3}) (x + 5)$

Raise $- 4$ to the power of $3$ .

$x^{2} (x^{3} - 12 x^{2} + 48 x - 64) (x + 5)$

Apply the distributive property.

$(x^{2} x^{3} + x^{2} (- 12 x^{2}) + x^{2} (48 x) + x^{2} \cdot - 64) (x + 5)$

Simplify.

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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

$(x^{2 + 3} + x^{2} (- 12 x^{2}) + x^{2} (48 x) + x^{2} \cdot - 64) (x + 5)$

Add $2$ and $3$ .

$(x^{5} + x^{2} (- 12 x^{2}) + x^{2} (48 x) + x^{2} \cdot - 64) (x + 5)$

Rewrite using the commutative property of multiplication.

$(x^{5} - 12 x^{2} x^{2} + x^{2} (48 x) + x^{2} \cdot - 64) (x + 5)$

Rewrite using the commutative property of multiplication.

$(x^{5} - 12 x^{2} x^{2} + 48 x^{2} x + x^{2} \cdot - 64) (x + 5)$

Move $- 64$ to the left of $x^{2}$ .

$(x^{5} - 12 x^{2} x^{2} + 48 x^{2} x - 64 \cdot x^{2}) (x + 5)$

Simplify each term.

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

Move $x^{2}$ .

$(x^{5} - 12 (x^{2} x^{2}) + 48 x^{2} x - 64 \cdot x^{2}) (x + 5)$

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

$(x^{5} - 12 x^{2 + 2} + 48 x^{2} x - 64 \cdot x^{2}) (x + 5)$

Add $2$ and $2$ .

$(x^{5} - 12 x^{4} + 48 x^{2} x - 64 \cdot x^{2}) (x + 5)$

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

Move $x$ .

$(x^{5} - 12 x^{4} + 48 (x \cdot x^{2}) - 64 \cdot x^{2}) (x + 5)$

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

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

$(x^{5} - 12 x^{4} + 48 (x^{1} x^{2}) - 64 \cdot x^{2}) (x + 5)$

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

$(x^{5} - 12 x^{4} + 48 x^{1 + 2} - 64 \cdot x^{2}) (x + 5)$

Add $1$ and $2$ .

$(x^{5} - 12 x^{4} + 48 x^{3} - 64 \cdot x^{2}) (x + 5)$

$(x^{5} - 12 x^{4} + 48 x^{3} - 64 x^{2}) (x + 5)$

Expand $(x^{5} - 12 x^{4} + 48 x^{3} - 64 x^{2}) (x + 5)$ by multiplying each term in the first expression by each term in the second expression.

$x^{5} x + x^{5} \cdot 5 - 12 x^{4} x - 12 x^{4} \cdot 5 + 48 x^{3} x + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

Simplify terms.

Simplify each term.

Multiply $x^{5}$ by $x$ by adding the exponents.

Multiply $x^{5}$ by $x$ .

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

$x^{5} x^{1} + x^{5} \cdot 5 - 12 x^{4} x - 12 x^{4} \cdot 5 + 48 x^{3} x + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

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

$x^{5 + 1} + x^{5} \cdot 5 - 12 x^{4} x - 12 x^{4} \cdot 5 + 48 x^{3} x + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

Add $5$ and $1$ .

$x^{6} + x^{5} \cdot 5 - 12 x^{4} x - 12 x^{4} \cdot 5 + 48 x^{3} x + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

Move $5$ to the left of $x^{5}$ .

$x^{6} + 5 \cdot x^{5} - 12 x^{4} x - 12 x^{4} \cdot 5 + 48 x^{3} x + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

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

Move $x$ .

$x^{6} + 5 x^{5} - 12 (x \cdot x^{4}) - 12 x^{4} \cdot 5 + 48 x^{3} x + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

Multiply $x$ by $x^{4}$ .

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

$x^{6} + 5 x^{5} - 12 (x^{1} x^{4}) - 12 x^{4} \cdot 5 + 48 x^{3} x + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

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

$x^{6} + 5 x^{5} - 12 x^{1 + 4} - 12 x^{4} \cdot 5 + 48 x^{3} x + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

Add $1$ and $4$ .

$x^{6} + 5 x^{5} - 12 x^{5} - 12 x^{4} \cdot 5 + 48 x^{3} x + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

Multiply $5$ by $- 12$ .

$x^{6} + 5 x^{5} - 12 x^{5} - 60 x^{4} + 48 x^{3} x + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

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

Move $x$ .

$x^{6} + 5 x^{5} - 12 x^{5} - 60 x^{4} + 48 (x \cdot x^{3}) + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

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

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

$x^{6} + 5 x^{5} - 12 x^{5} - 60 x^{4} + 48 (x^{1} x^{3}) + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

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

$x^{6} + 5 x^{5} - 12 x^{5} - 60 x^{4} + 48 x^{1 + 3} + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

Add $1$ and $3$ .

$x^{6} + 5 x^{5} - 12 x^{5} - 60 x^{4} + 48 x^{4} + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

Multiply $5$ by $48$ .

$x^{6} + 5 x^{5} - 12 x^{5} - 60 x^{4} + 48 x^{4} + 240 x^{3} - 64 x^{2} x - 64 x^{2} \cdot 5$

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

Move $x$ .

$x^{6} + 5 x^{5} - 12 x^{5} - 60 x^{4} + 48 x^{4} + 240 x^{3} - 64 (x \cdot x^{2}) - 64 x^{2} \cdot 5$

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

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

$x^{6} + 5 x^{5} - 12 x^{5} - 60 x^{4} + 48 x^{4} + 240 x^{3} - 64 (x^{1} x^{2}) - 64 x^{2} \cdot 5$

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

$x^{6} + 5 x^{5} - 12 x^{5} - 60 x^{4} + 48 x^{4} + 240 x^{3} - 64 x^{1 + 2} - 64 x^{2} \cdot 5$

Add $1$ and $2$ .

$x^{6} + 5 x^{5} - 12 x^{5} - 60 x^{4} + 48 x^{4} + 240 x^{3} - 64 x^{3} - 64 x^{2} \cdot 5$

Multiply $5$ by $- 64$ .

$x^{6} + 5 x^{5} - 12 x^{5} - 60 x^{4} + 48 x^{4} + 240 x^{3} - 64 x^{3} - 320 x^{2}$

Simplify by adding terms.

Subtract $12 x^{5}$ from $5 x^{5}$ .

$x^{6} - 7 x^{5} - 60 x^{4} + 48 x^{4} + 240 x^{3} - 64 x^{3} - 320 x^{2}$

Add $- 60 x^{4}$ and $48 x^{4}$ .

$x^{6} - 7 x^{5} - 12 x^{4} + 240 x^{3} - 64 x^{3} - 320 x^{2}$

Subtract $64 x^{3}$ from $240 x^{3}$ .

$x^{6} - 7 x^{5} - 12 x^{4} + 176 x^{3} - 320 x^{2}$

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.

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

Use the Binomial Theorem.

$x^{2} (x^{3} + 3 x^{2} \cdot - 4 + 3 x {(- 4)}^{2} + {(- 4)}^{3}) (x + 5)$

Simplify terms.

Simplify each term.

Multiply $- 4$ by $3$ .

$x^{2} (x^{3} - 12 x^{2} + 3 x {(- 4)}^{2} + {(- 4)}^{3}) (x + 5)$

Raise $- 4$ to the power of $2$ .

$x^{2} (x^{3} - 12 x^{2} + 3 x \cdot 16 + {(- 4)}^{3}) (x + 5)$

Multiply $16$ by $3$ .

$x^{2} (x^{3} - 12 x^{2} + 48 x + {(- 4)}^{3}) (x + 5)$

Raise $- 4$ to the power of $3$ .

$x^{2} (x^{3} - 12 x^{2} + 48 x - 64) (x + 5)$

Apply the distributive property.

$(x^{2} x^{3} + x^{2} (- 12 x^{2}) + x^{2} (48 x) + x^{2} \cdot - 64) (x + 5)$

Simplify.

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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

$(x^{2 + 3} + x^{2} (- 12 x^{2}) + x^{2} (48 x) + x^{2} \cdot - 64) (x + 5)$

Add $2$ and $3$ .

$(x^{5} + x^{2} (- 12 x^{2}) + x^{2} (48 x) + x^{2} \cdot - 64) (x + 5)$

Rewrite using the commutative property of multiplication.

$(x^{5} - 12 x^{2} x^{2} + x^{2} (48 x) + x^{2} \cdot - 64) (x + 5)$

Rewrite using the commutative property of multiplication.

$(x^{5} - 12 x^{2} x^{2} + 48 x^{2} x + x^{2} \cdot - 64) (x + 5)$

Move $- 64$ to the left of $x^{2}$ .

$(x^{5} - 12 x^{2} x^{2} + 48 x^{2} x - 64 \cdot x^{2}) (x + 5)$

Simplify each term.

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

Move $x^{2}$ .

$(x^{5} - 12 (x^{2} x^{2}) + 48 x^{2} x - 64 \cdot x^{2}) (x + 5)$

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

$(x^{5} - 12 x^{2 + 2} + 48 x^{2} x - 64 \cdot x^{2}) (x + 5)$

Add $2$ and $2$ .

$(x^{5} - 12 x^{4} + 48 x^{2} x - 64 \cdot x^{2}) (x + 5)$

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

Move $x$ .

$(x^{5} - 12 x^{4} + 48 (x \cdot x^{2}) - 64 \cdot x^{2}) (x + 5)$

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

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

$(x^{5} - 12 x^{4} + 48 (x^{1} x^{2}) - 64 \cdot x^{2}) (x + 5)$

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

$(x^{5} - 12 x^{4} + 48 x^{1 + 2} - 64 \cdot x^{2}) (x + 5)$

Add $1$ and $2$ .

$(x^{5} - 12 x^{4} + 48 x^{3} - 64 \cdot x^{2}) (x + 5)$

$(x^{5} - 12 x^{4} + 48 x^{3} - 64 x^{2}) (x + 5)$

Expand $(x^{5} - 12 x^{4} + 48 x^{3} - 64 x^{2}) (x + 5)$ by multiplying each term in the first expression by each term in the second expression.

$x^{5} x + x^{5} \cdot 5 - 12 x^{4} x - 12 x^{4} \cdot 5 + 48 x^{3} x + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

Simplify terms.

Simplify each term.

Multiply $x^{5}$ by $x$ by adding the exponents.

Multiply $x^{5}$ by $x$ .

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

$x^{5} x^{1} + x^{5} \cdot 5 - 12 x^{4} x - 12 x^{4} \cdot 5 + 48 x^{3} x + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

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

$x^{5 + 1} + x^{5} \cdot 5 - 12 x^{4} x - 12 x^{4} \cdot 5 + 48 x^{3} x + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

Add $5$ and $1$ .

$x^{6} + x^{5} \cdot 5 - 12 x^{4} x - 12 x^{4} \cdot 5 + 48 x^{3} x + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

Move $5$ to the left of $x^{5}$ .

$x^{6} + 5 \cdot x^{5} - 12 x^{4} x - 12 x^{4} \cdot 5 + 48 x^{3} x + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

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

Move $x$ .

$x^{6} + 5 x^{5} - 12 (x \cdot x^{4}) - 12 x^{4} \cdot 5 + 48 x^{3} x + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

Multiply $x$ by $x^{4}$ .

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

$x^{6} + 5 x^{5} - 12 (x^{1} x^{4}) - 12 x^{4} \cdot 5 + 48 x^{3} x + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

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

$x^{6} + 5 x^{5} - 12 x^{1 + 4} - 12 x^{4} \cdot 5 + 48 x^{3} x + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

Add $1$ and $4$ .

$x^{6} + 5 x^{5} - 12 x^{5} - 12 x^{4} \cdot 5 + 48 x^{3} x + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

Multiply $5$ by $- 12$ .

$x^{6} + 5 x^{5} - 12 x^{5} - 60 x^{4} + 48 x^{3} x + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

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

Move $x$ .

$x^{6} + 5 x^{5} - 12 x^{5} - 60 x^{4} + 48 (x \cdot x^{3}) + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

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

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

$x^{6} + 5 x^{5} - 12 x^{5} - 60 x^{4} + 48 (x^{1} x^{3}) + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

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

$x^{6} + 5 x^{5} - 12 x^{5} - 60 x^{4} + 48 x^{1 + 3} + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

Add $1$ and $3$ .

$x^{6} + 5 x^{5} - 12 x^{5} - 60 x^{4} + 48 x^{4} + 48 x^{3} \cdot 5 - 64 x^{2} x - 64 x^{2} \cdot 5$

Multiply $5$ by $48$ .

$x^{6} + 5 x^{5} - 12 x^{5} - 60 x^{4} + 48 x^{4} + 240 x^{3} - 64 x^{2} x - 64 x^{2} \cdot 5$

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

Move $x$ .

$x^{6} + 5 x^{5} - 12 x^{5} - 60 x^{4} + 48 x^{4} + 240 x^{3} - 64 (x \cdot x^{2}) - 64 x^{2} \cdot 5$

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

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

$x^{6} + 5 x^{5} - 12 x^{5} - 60 x^{4} + 48 x^{4} + 240 x^{3} - 64 (x^{1} x^{2}) - 64 x^{2} \cdot 5$

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

$x^{6} + 5 x^{5} - 12 x^{5} - 60 x^{4} + 48 x^{4} + 240 x^{3} - 64 x^{1 + 2} - 64 x^{2} \cdot 5$

Add $1$ and $2$ .

$x^{6} + 5 x^{5} - 12 x^{5} - 60 x^{4} + 48 x^{4} + 240 x^{3} - 64 x^{3} - 64 x^{2} \cdot 5$

Multiply $5$ by $- 64$ .

$x^{6} + 5 x^{5} - 12 x^{5} - 60 x^{4} + 48 x^{4} + 240 x^{3} - 64 x^{3} - 320 x^{2}$

Simplify by adding terms.

Subtract $12 x^{5}$ from $5 x^{5}$ .

$x^{6} - 7 x^{5} - 60 x^{4} + 48 x^{4} + 240 x^{3} - 64 x^{3} - 320 x^{2}$

Add $- 60 x^{4}$ and $48 x^{4}$ .

$x^{6} - 7 x^{5} - 12 x^{4} + 240 x^{3} - 64 x^{3} - 320 x^{2}$

Subtract $64 x^{3}$ from $240 x^{3}$ .

$x^{6} - 7 x^{5} - 12 x^{4} + 176 x^{3} - 320 x^{2}$

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

$x^{6}$

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

$1$

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.

Rises to the left and rises to the right

Step 7

Find the Behavior (Leading Coefficient Test) f(x)=x^2(x-4)^3(x+5)

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