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

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

Step 1

Write ${(x^{2} + 1)}^{2} (2 x - 3)$ as a function.

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

Step 2

Identify the degree of the function.

Simplify and reorder the polynomial.

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

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

Expand $(x^{2} + 1) (x^{2} + 1)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

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

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

Add $2$ and $2$ .

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

Multiply $x^{2}$ by $1$ .

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

Multiply $x^{2}$ by $1$ .

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

Multiply $1$ by $1$ .

$(x^{4} + x^{2} + x^{2} + 1) (2 x - 3)$

Add $x^{2}$ and $x^{2}$ .

$(x^{4} + 2 x^{2} + 1) (2 x - 3)$

Expand $(x^{4} + 2 x^{2} + 1) (2 x - 3)$ by multiplying each term in the first expression by each term in the second expression.

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

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $4$ .

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

Move $- 3$ to the left of $x^{4}$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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

Multiply $2$ by $2$ .

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

Multiply $- 3$ by $2$ .

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

Multiply $2 x$ by $1$ .

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

Multiply $- 3$ by $1$ .

$2 x^{5} - 3 x^{4} + 4 x^{3} - 6 x^{2} + 2 x - 3$

The largest exponent is the degree of the polynomial.

$5$

Step 3

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

Odd

Step 4

Identify the leading coefficient.

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

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

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

Expand $(x^{2} + 1) (x^{2} + 1)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

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

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

Add $2$ and $2$ .

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

Multiply $x^{2}$ by $1$ .

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

Multiply $x^{2}$ by $1$ .

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

Multiply $1$ by $1$ .

$(x^{4} + x^{2} + x^{2} + 1) (2 x - 3)$

Add $x^{2}$ and $x^{2}$ .

$(x^{4} + 2 x^{2} + 1) (2 x - 3)$

Expand $(x^{4} + 2 x^{2} + 1) (2 x - 3)$ by multiplying each term in the first expression by each term in the second expression.

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

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $4$ .

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

Move $- 3$ to the left of $x^{4}$ .

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

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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

Multiply $2$ by $2$ .

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

Multiply $- 3$ by $2$ .

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

Multiply $2 x$ by $1$ .

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

Multiply $- 3$ by $1$ .

$2 x^{5} - 3 x^{4} + 4 x^{3} - 6 x^{2} + 2 x - 3$

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

$2 x^{5}$

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

$2$

Step 5

Since the leading coefficient is positive, the graph rises to the right.

Positive

Step 6

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 7

Determine the behavior.

Falls to the left and rises to the right

Step 8

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

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