Find the Behavior (Leading Coefficient Test) f(x)=(x+6)(2x-3)(x-1)^2

$f (x) = (x + 6) (2 x - 3) {(x - 1)}^{2}$

Step 1

Identify the degree of the function.

Simplify and reorder the polynomial.

Expand $(x + 6) (2 x - 3)$ using the FOIL Method.

Apply the distributive property.

$(x (2 x - 3) + 6 (2 x - 3)) {(x - 1)}^{2}$

Apply the distributive property.

$(x (2 x) + x \cdot - 3 + 6 (2 x - 3)) {(x - 1)}^{2}$

Apply the distributive property.

$(x (2 x) + x \cdot - 3 + 6 (2 x) + 6 \cdot - 3) {(x - 1)}^{2}$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$(2 x \cdot x + x \cdot - 3 + 6 (2 x) + 6 \cdot - 3) {(x - 1)}^{2}$

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

Move $x$ .

$(2 (x \cdot x) + x \cdot - 3 + 6 (2 x) + 6 \cdot - 3) {(x - 1)}^{2}$

Multiply $x$ by $x$ .

$(2 x^{2} + x \cdot - 3 + 6 (2 x) + 6 \cdot - 3) {(x - 1)}^{2}$

Move $- 3$ to the left of $x$ .

$(2 x^{2} - 3 \cdot x + 6 (2 x) + 6 \cdot - 3) {(x - 1)}^{2}$

Multiply $2$ by $6$ .

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

Multiply $6$ by $- 3$ .

$(2 x^{2} - 3 x + 12 x - 18) {(x - 1)}^{2}$

Add $- 3 x$ and $12 x$ .

$(2 x^{2} + 9 x - 18) {(x - 1)}^{2}$

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$(2 x^{2} + 9 x - 18) ((x - 1) (x - 1))$

Expand $(x - 1) (x - 1)$ using the FOIL Method.

Apply the distributive property.

$(2 x^{2} + 9 x - 18) (x (x - 1) - 1 (x - 1))$

Apply the distributive property.

$(2 x^{2} + 9 x - 18) (x \cdot x + x \cdot - 1 - 1 (x - 1))$

Apply the distributive property.

$(2 x^{2} + 9 x - 18) (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$(2 x^{2} + 9 x - 18) (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1)$

Move $- 1$ to the left of $x$ .

$(2 x^{2} + 9 x - 18) (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1)$

Rewrite $- 1 x$ as $- x$ .

$(2 x^{2} + 9 x - 18) (x^{2} - x - 1 x - 1 \cdot - 1)$

Rewrite $- 1 x$ as $- x$ .

$(2 x^{2} + 9 x - 18) (x^{2} - x - x - 1 \cdot - 1)$

Multiply $- 1$ by $- 1$ .

$(2 x^{2} + 9 x - 18) (x^{2} - x - x + 1)$

Subtract $x$ from $- x$ .

$(2 x^{2} + 9 x - 18) (x^{2} - 2 x + 1)$

Expand $(2 x^{2} + 9 x - 18) (x^{2} - 2 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$2 x^{2} x^{2} + 2 x^{2} (- 2 x) + 2 x^{2} \cdot 1 + 9 x \cdot x^{2} + 9 x (- 2 x) + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

Simplify terms.

Simplify each term.

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

Move $x^{2}$ .

$2 (x^{2} x^{2}) + 2 x^{2} (- 2 x) + 2 x^{2} \cdot 1 + 9 x \cdot x^{2} + 9 x (- 2 x) + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

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

$2 x^{2 + 2} + 2 x^{2} (- 2 x) + 2 x^{2} \cdot 1 + 9 x \cdot x^{2} + 9 x (- 2 x) + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

Add $2$ and $2$ .

$2 x^{4} + 2 x^{2} (- 2 x) + 2 x^{2} \cdot 1 + 9 x \cdot x^{2} + 9 x (- 2 x) + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

Rewrite using the commutative property of multiplication.

$2 x^{4} + 2 \cdot - 2 x^{2} x + 2 x^{2} \cdot 1 + 9 x \cdot x^{2} + 9 x (- 2 x) + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

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

Move $x$ .

$2 x^{4} + 2 \cdot - 2 (x \cdot x^{2}) + 2 x^{2} \cdot 1 + 9 x \cdot x^{2} + 9 x (- 2 x) + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

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

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

$2 x^{4} + 2 \cdot - 2 (x^{1} x^{2}) + 2 x^{2} \cdot 1 + 9 x \cdot x^{2} + 9 x (- 2 x) + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

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

$2 x^{4} + 2 \cdot - 2 x^{1 + 2} + 2 x^{2} \cdot 1 + 9 x \cdot x^{2} + 9 x (- 2 x) + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

Add $1$ and $2$ .

$2 x^{4} + 2 \cdot - 2 x^{3} + 2 x^{2} \cdot 1 + 9 x \cdot x^{2} + 9 x (- 2 x) + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

Multiply $2$ by $- 2$ .

$2 x^{4} - 4 x^{3} + 2 x^{2} \cdot 1 + 9 x \cdot x^{2} + 9 x (- 2 x) + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

Multiply $2$ by $1$ .

$2 x^{4} - 4 x^{3} + 2 x^{2} + 9 x \cdot x^{2} + 9 x (- 2 x) + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

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

Move $x^{2}$ .

$2 x^{4} - 4 x^{3} + 2 x^{2} + 9 (x^{2} x) + 9 x (- 2 x) + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

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

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

$2 x^{4} - 4 x^{3} + 2 x^{2} + 9 (x^{2} x^{1}) + 9 x (- 2 x) + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

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

$2 x^{4} - 4 x^{3} + 2 x^{2} + 9 x^{2 + 1} + 9 x (- 2 x) + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

Add $2$ and $1$ .

$2 x^{4} - 4 x^{3} + 2 x^{2} + 9 x^{3} + 9 x (- 2 x) + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

Rewrite using the commutative property of multiplication.

$2 x^{4} - 4 x^{3} + 2 x^{2} + 9 x^{3} + 9 \cdot - 2 x \cdot x + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

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

Move $x$ .

$2 x^{4} - 4 x^{3} + 2 x^{2} + 9 x^{3} + 9 \cdot - 2 (x \cdot x) + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

Multiply $x$ by $x$ .

$2 x^{4} - 4 x^{3} + 2 x^{2} + 9 x^{3} + 9 \cdot - 2 x^{2} + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

Multiply $9$ by $- 2$ .

$2 x^{4} - 4 x^{3} + 2 x^{2} + 9 x^{3} - 18 x^{2} + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

Multiply $9$ by $1$ .

$2 x^{4} - 4 x^{3} + 2 x^{2} + 9 x^{3} - 18 x^{2} + 9 x - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

Multiply $- 2$ by $- 18$ .

$2 x^{4} - 4 x^{3} + 2 x^{2} + 9 x^{3} - 18 x^{2} + 9 x - 18 x^{2} + 36 x - 18 \cdot 1$

Multiply $- 18$ by $1$ .

$2 x^{4} - 4 x^{3} + 2 x^{2} + 9 x^{3} - 18 x^{2} + 9 x - 18 x^{2} + 36 x - 18$

Simplify by adding terms.

Add $- 4 x^{3}$ and $9 x^{3}$ .

$2 x^{4} + 5 x^{3} + 2 x^{2} - 18 x^{2} + 9 x - 18 x^{2} + 36 x - 18$

Subtract $18 x^{2}$ from $2 x^{2}$ .

$2 x^{4} + 5 x^{3} - 16 x^{2} + 9 x - 18 x^{2} + 36 x - 18$

Subtract $18 x^{2}$ from $- 16 x^{2}$ .

$2 x^{4} + 5 x^{3} - 34 x^{2} + 9 x + 36 x - 18$

Add $9 x$ and $36 x$ .

$2 x^{4} + 5 x^{3} - 34 x^{2} + 45 x - 18$

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.

Expand $(x + 6) (2 x - 3)$ using the FOIL Method.

Apply the distributive property.

$(x (2 x - 3) + 6 (2 x - 3)) {(x - 1)}^{2}$

Apply the distributive property.

$(x (2 x) + x \cdot - 3 + 6 (2 x - 3)) {(x - 1)}^{2}$

Apply the distributive property.

$(x (2 x) + x \cdot - 3 + 6 (2 x) + 6 \cdot - 3) {(x - 1)}^{2}$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$(2 x \cdot x + x \cdot - 3 + 6 (2 x) + 6 \cdot - 3) {(x - 1)}^{2}$

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

Move $x$ .

$(2 (x \cdot x) + x \cdot - 3 + 6 (2 x) + 6 \cdot - 3) {(x - 1)}^{2}$

Multiply $x$ by $x$ .

$(2 x^{2} + x \cdot - 3 + 6 (2 x) + 6 \cdot - 3) {(x - 1)}^{2}$

Move $- 3$ to the left of $x$ .

$(2 x^{2} - 3 \cdot x + 6 (2 x) + 6 \cdot - 3) {(x - 1)}^{2}$

Multiply $2$ by $6$ .

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

Multiply $6$ by $- 3$ .

$(2 x^{2} - 3 x + 12 x - 18) {(x - 1)}^{2}$

Add $- 3 x$ and $12 x$ .

$(2 x^{2} + 9 x - 18) {(x - 1)}^{2}$

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$(2 x^{2} + 9 x - 18) ((x - 1) (x - 1))$

Expand $(x - 1) (x - 1)$ using the FOIL Method.

Apply the distributive property.

$(2 x^{2} + 9 x - 18) (x (x - 1) - 1 (x - 1))$

Apply the distributive property.

$(2 x^{2} + 9 x - 18) (x \cdot x + x \cdot - 1 - 1 (x - 1))$

Apply the distributive property.

$(2 x^{2} + 9 x - 18) (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$(2 x^{2} + 9 x - 18) (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1)$

Move $- 1$ to the left of $x$ .

$(2 x^{2} + 9 x - 18) (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1)$

Rewrite $- 1 x$ as $- x$ .

$(2 x^{2} + 9 x - 18) (x^{2} - x - 1 x - 1 \cdot - 1)$

Rewrite $- 1 x$ as $- x$ .

$(2 x^{2} + 9 x - 18) (x^{2} - x - x - 1 \cdot - 1)$

Multiply $- 1$ by $- 1$ .

$(2 x^{2} + 9 x - 18) (x^{2} - x - x + 1)$

Subtract $x$ from $- x$ .

$(2 x^{2} + 9 x - 18) (x^{2} - 2 x + 1)$

Expand $(2 x^{2} + 9 x - 18) (x^{2} - 2 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$2 x^{2} x^{2} + 2 x^{2} (- 2 x) + 2 x^{2} \cdot 1 + 9 x \cdot x^{2} + 9 x (- 2 x) + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

Simplify terms.

Simplify each term.

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

Move $x^{2}$ .

$2 (x^{2} x^{2}) + 2 x^{2} (- 2 x) + 2 x^{2} \cdot 1 + 9 x \cdot x^{2} + 9 x (- 2 x) + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

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

$2 x^{2 + 2} + 2 x^{2} (- 2 x) + 2 x^{2} \cdot 1 + 9 x \cdot x^{2} + 9 x (- 2 x) + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

Add $2$ and $2$ .

$2 x^{4} + 2 x^{2} (- 2 x) + 2 x^{2} \cdot 1 + 9 x \cdot x^{2} + 9 x (- 2 x) + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

Rewrite using the commutative property of multiplication.

$2 x^{4} + 2 \cdot - 2 x^{2} x + 2 x^{2} \cdot 1 + 9 x \cdot x^{2} + 9 x (- 2 x) + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

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

Move $x$ .

$2 x^{4} + 2 \cdot - 2 (x \cdot x^{2}) + 2 x^{2} \cdot 1 + 9 x \cdot x^{2} + 9 x (- 2 x) + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

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

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

$2 x^{4} + 2 \cdot - 2 (x^{1} x^{2}) + 2 x^{2} \cdot 1 + 9 x \cdot x^{2} + 9 x (- 2 x) + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

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

$2 x^{4} + 2 \cdot - 2 x^{1 + 2} + 2 x^{2} \cdot 1 + 9 x \cdot x^{2} + 9 x (- 2 x) + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

Add $1$ and $2$ .

$2 x^{4} + 2 \cdot - 2 x^{3} + 2 x^{2} \cdot 1 + 9 x \cdot x^{2} + 9 x (- 2 x) + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

Multiply $2$ by $- 2$ .

$2 x^{4} - 4 x^{3} + 2 x^{2} \cdot 1 + 9 x \cdot x^{2} + 9 x (- 2 x) + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

Multiply $2$ by $1$ .

$2 x^{4} - 4 x^{3} + 2 x^{2} + 9 x \cdot x^{2} + 9 x (- 2 x) + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

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

Move $x^{2}$ .

$2 x^{4} - 4 x^{3} + 2 x^{2} + 9 (x^{2} x) + 9 x (- 2 x) + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

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

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

$2 x^{4} - 4 x^{3} + 2 x^{2} + 9 (x^{2} x^{1}) + 9 x (- 2 x) + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

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

$2 x^{4} - 4 x^{3} + 2 x^{2} + 9 x^{2 + 1} + 9 x (- 2 x) + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

Add $2$ and $1$ .

$2 x^{4} - 4 x^{3} + 2 x^{2} + 9 x^{3} + 9 x (- 2 x) + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

Rewrite using the commutative property of multiplication.

$2 x^{4} - 4 x^{3} + 2 x^{2} + 9 x^{3} + 9 \cdot - 2 x \cdot x + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

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

Move $x$ .

$2 x^{4} - 4 x^{3} + 2 x^{2} + 9 x^{3} + 9 \cdot - 2 (x \cdot x) + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

Multiply $x$ by $x$ .

$2 x^{4} - 4 x^{3} + 2 x^{2} + 9 x^{3} + 9 \cdot - 2 x^{2} + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

Multiply $9$ by $- 2$ .

$2 x^{4} - 4 x^{3} + 2 x^{2} + 9 x^{3} - 18 x^{2} + 9 x \cdot 1 - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

Multiply $9$ by $1$ .

$2 x^{4} - 4 x^{3} + 2 x^{2} + 9 x^{3} - 18 x^{2} + 9 x - 18 x^{2} - 18 (- 2 x) - 18 \cdot 1$

Multiply $- 2$ by $- 18$ .

$2 x^{4} - 4 x^{3} + 2 x^{2} + 9 x^{3} - 18 x^{2} + 9 x - 18 x^{2} + 36 x - 18 \cdot 1$

Multiply $- 18$ by $1$ .

$2 x^{4} - 4 x^{3} + 2 x^{2} + 9 x^{3} - 18 x^{2} + 9 x - 18 x^{2} + 36 x - 18$

Simplify by adding terms.

Add $- 4 x^{3}$ and $9 x^{3}$ .

$2 x^{4} + 5 x^{3} + 2 x^{2} - 18 x^{2} + 9 x - 18 x^{2} + 36 x - 18$

Subtract $18 x^{2}$ from $2 x^{2}$ .

$2 x^{4} + 5 x^{3} - 16 x^{2} + 9 x - 18 x^{2} + 36 x - 18$

Subtract $18 x^{2}$ from $- 16 x^{2}$ .

$2 x^{4} + 5 x^{3} - 34 x^{2} + 9 x + 36 x - 18$

Add $9 x$ and $36 x$ .

$2 x^{4} + 5 x^{3} - 34 x^{2} + 45 x - 18$

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

$2 x^{4}$

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

$2$

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.

Rises to the left and rises to the right

Step 7

Find the Behavior (Leading Coefficient Test) f(x)=(x+6)(2x-3)(x-1)^2

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