Find the Symmetry f(x)=x^2(x-3)^3(x+4)

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

Step 1

Determine if the function is odd, even, or neither in order to find the symmetry.

1. If odd, the function is symmetric about the origin.

2. If even, the function is symmetric about the y-axis.

Step 2

Simplify.

Use the Binomial Theorem.

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

Simplify terms.

Simplify each term.

Multiply $- 3$ by $3$ .

$f (x) = x^{2} (x^{3} - 9 x^{2} + 3 x {(- 3)}^{2} + {(- 3)}^{3}) (x + 4)$

Raise $- 3$ to the power of $2$ .

$f (x) = x^{2} (x^{3} - 9 x^{2} + 3 x \cdot 9 + {(- 3)}^{3}) (x + 4)$

Multiply $9$ by $3$ .

$f (x) = x^{2} (x^{3} - 9 x^{2} + 27 x + {(- 3)}^{3}) (x + 4)$

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

$f (x) = x^{2} (x^{3} - 9 x^{2} + 27 x - 27) (x + 4)$

Apply the distributive property.

$f (x) = (x^{2} x^{3} + x^{2} (- 9 x^{2}) + x^{2} (27 x) + x^{2} \cdot - 27) (x + 4)$

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.

$f (x) = (x^{2 + 3} + x^{2} (- 9 x^{2}) + x^{2} (27 x) + x^{2} \cdot - 27) (x + 4)$

Add $2$ and $3$ .

$f (x) = (x^{5} + x^{2} (- 9 x^{2}) + x^{2} (27 x) + x^{2} \cdot - 27) (x + 4)$

Rewrite using the commutative property of multiplication.

$f (x) = (x^{5} - 9 x^{2} x^{2} + x^{2} (27 x) + x^{2} \cdot - 27) (x + 4)$

Rewrite using the commutative property of multiplication.

$f (x) = (x^{5} - 9 x^{2} x^{2} + 27 x^{2} x + x^{2} \cdot - 27) (x + 4)$

Move $- 27$ to the left of $x^{2}$ .

$f (x) = (x^{5} - 9 x^{2} x^{2} + 27 x^{2} x - 27 \cdot x^{2}) (x + 4)$

Simplify each term.

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

Move $x^{2}$ .

$f (x) = (x^{5} - 9 (x^{2} x^{2}) + 27 x^{2} x - 27 \cdot x^{2}) (x + 4)$

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

$f (x) = (x^{5} - 9 x^{2 + 2} + 27 x^{2} x - 27 \cdot x^{2}) (x + 4)$

Add $2$ and $2$ .

$f (x) = (x^{5} - 9 x^{4} + 27 x^{2} x - 27 \cdot x^{2}) (x + 4)$

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

Move $x$ .

$f (x) = (x^{5} - 9 x^{4} + 27 (x \cdot x^{2}) - 27 \cdot x^{2}) (x + 4)$

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

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

$f (x) = (x^{5} - 9 x^{4} + 27 (x \cdot x^{2}) - 27 \cdot x^{2}) (x + 4)$

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

$f (x) = (x^{5} - 9 x^{4} + 27 x^{1 + 2} - 27 \cdot x^{2}) (x + 4)$

Add $1$ and $2$ .

$f (x) = (x^{5} - 9 x^{4} + 27 x^{3} - 27 \cdot x^{2}) (x + 4)$

$f (x) = (x^{5} - 9 x^{4} + 27 x^{3} - 27 x^{2}) (x + 4)$

Expand $(x^{5} - 9 x^{4} + 27 x^{3} - 27 x^{2}) (x + 4)$ by multiplying each term in the first expression by each term in the second expression.

$f (x) = x^{5} x + x^{5} \cdot 4 - 9 x^{4} x - 9 x^{4} \cdot 4 + 27 x^{3} x + 27 x^{3} \cdot 4 - 27 x^{2} x - 27 x^{2} \cdot 4$

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 $1$ .

$f (x) = x^{5} x + x^{5} \cdot 4 - 9 x^{4} x - 9 x^{4} \cdot 4 + 27 x^{3} x + 27 x^{3} \cdot 4 - 27 x^{2} x - 27 x^{2} \cdot 4$

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

$f (x) = x^{5 + 1} + x^{5} \cdot 4 - 9 x^{4} x - 9 x^{4} \cdot 4 + 27 x^{3} x + 27 x^{3} \cdot 4 - 27 x^{2} x - 27 x^{2} \cdot 4$

Add $5$ and $1$ .

$f (x) = x^{6} + x^{5} \cdot 4 - 9 x^{4} x - 9 x^{4} \cdot 4 + 27 x^{3} x + 27 x^{3} \cdot 4 - 27 x^{2} x - 27 x^{2} \cdot 4$

Move $4$ to the left of $x^{5}$ .

$f (x) = x^{6} + 4 \cdot x^{5} - 9 x^{4} x - 9 x^{4} \cdot 4 + 27 x^{3} x + 27 x^{3} \cdot 4 - 27 x^{2} x - 27 x^{2} \cdot 4$

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

Move $x$ .

$f (x) = x^{6} + 4 x^{5} - 9 (x \cdot x^{4}) - 9 x^{4} \cdot 4 + 27 x^{3} x + 27 x^{3} \cdot 4 - 27 x^{2} x - 27 x^{2} \cdot 4$

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

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

$f (x) = x^{6} + 4 x^{5} - 9 (x \cdot x^{4}) - 9 x^{4} \cdot 4 + 27 x^{3} x + 27 x^{3} \cdot 4 - 27 x^{2} x - 27 x^{2} \cdot 4$

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

$f (x) = x^{6} + 4 x^{5} - 9 x^{1 + 4} - 9 x^{4} \cdot 4 + 27 x^{3} x + 27 x^{3} \cdot 4 - 27 x^{2} x - 27 x^{2} \cdot 4$

Add $1$ and $4$ .

$f (x) = x^{6} + 4 x^{5} - 9 x^{5} - 9 x^{4} \cdot 4 + 27 x^{3} x + 27 x^{3} \cdot 4 - 27 x^{2} x - 27 x^{2} \cdot 4$

Multiply $4$ by $- 9$ .

$f (x) = x^{6} + 4 x^{5} - 9 x^{5} - 36 x^{4} + 27 x^{3} x + 27 x^{3} \cdot 4 - 27 x^{2} x - 27 x^{2} \cdot 4$

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

Move $x$ .

$f (x) = x^{6} + 4 x^{5} - 9 x^{5} - 36 x^{4} + 27 (x \cdot x^{3}) + 27 x^{3} \cdot 4 - 27 x^{2} x - 27 x^{2} \cdot 4$

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

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

$f (x) = x^{6} + 4 x^{5} - 9 x^{5} - 36 x^{4} + 27 (x \cdot x^{3}) + 27 x^{3} \cdot 4 - 27 x^{2} x - 27 x^{2} \cdot 4$

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

$f (x) = x^{6} + 4 x^{5} - 9 x^{5} - 36 x^{4} + 27 x^{1 + 3} + 27 x^{3} \cdot 4 - 27 x^{2} x - 27 x^{2} \cdot 4$

Add $1$ and $3$ .

$f (x) = x^{6} + 4 x^{5} - 9 x^{5} - 36 x^{4} + 27 x^{4} + 27 x^{3} \cdot 4 - 27 x^{2} x - 27 x^{2} \cdot 4$

Multiply $4$ by $27$ .

$f (x) = x^{6} + 4 x^{5} - 9 x^{5} - 36 x^{4} + 27 x^{4} + 108 x^{3} - 27 x^{2} x - 27 x^{2} \cdot 4$

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

Move $x$ .

$f (x) = x^{6} + 4 x^{5} - 9 x^{5} - 36 x^{4} + 27 x^{4} + 108 x^{3} - 27 (x \cdot x^{2}) - 27 x^{2} \cdot 4$

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

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

$f (x) = x^{6} + 4 x^{5} - 9 x^{5} - 36 x^{4} + 27 x^{4} + 108 x^{3} - 27 (x \cdot x^{2}) - 27 x^{2} \cdot 4$

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

$f (x) = x^{6} + 4 x^{5} - 9 x^{5} - 36 x^{4} + 27 x^{4} + 108 x^{3} - 27 x^{1 + 2} - 27 x^{2} \cdot 4$

Add $1$ and $2$ .

$f (x) = x^{6} + 4 x^{5} - 9 x^{5} - 36 x^{4} + 27 x^{4} + 108 x^{3} - 27 x^{3} - 27 x^{2} \cdot 4$

Multiply $4$ by $- 27$ .

$f (x) = x^{6} + 4 x^{5} - 9 x^{5} - 36 x^{4} + 27 x^{4} + 108 x^{3} - 27 x^{3} - 108 x^{2}$

Simplify by adding terms.

Subtract $9 x^{5}$ from $4 x^{5}$ .

$f (x) = x^{6} - 5 x^{5} - 36 x^{4} + 27 x^{4} + 108 x^{3} - 27 x^{3} - 108 x^{2}$

Add $- 36 x^{4}$ and $27 x^{4}$ .

$f (x) = x^{6} - 5 x^{5} - 9 x^{4} + 108 x^{3} - 27 x^{3} - 108 x^{2}$

Subtract $27 x^{3}$ from $108 x^{3}$ .

$f (x) = x^{6} - 5 x^{5} - 9 x^{4} + 81 x^{3} - 108 x^{2}$

Step 3

Find $f (- x)$ .

Find $f (- x)$ by substituting $- x$ for all occurrence of $x$ in $f (x)$ .

$f (- x) = {(- x)}^{6} - 5 {(- x)}^{5} - 9 {(- x)}^{4} + 81 {(- x)}^{3} - 108 {(- x)}^{2}$

Simplify each term.

Apply the product rule to $- x$ .

$f (- x) = {(- 1)}^{6} x^{6} - 5 {(- x)}^{5} - 9 {(- x)}^{4} + 81 {(- x)}^{3} - 108 {(- x)}^{2}$

Raise $- 1$ to the power of $6$ .

$f (- x) = 1 x^{6} - 5 {(- x)}^{5} - 9 {(- x)}^{4} + 81 {(- x)}^{3} - 108 {(- x)}^{2}$

Multiply $x^{6}$ by $1$ .

$f (- x) = x^{6} - 5 {(- x)}^{5} - 9 {(- x)}^{4} + 81 {(- x)}^{3} - 108 {(- x)}^{2}$

Apply the product rule to $- x$ .

$f (- x) = x^{6} - 5 ({(- 1)}^{5} x^{5}) - 9 {(- x)}^{4} + 81 {(- x)}^{3} - 108 {(- x)}^{2}$

Raise $- 1$ to the power of $5$ .

$f (- x) = x^{6} - 5 (- x^{5}) - 9 {(- x)}^{4} + 81 {(- x)}^{3} - 108 {(- x)}^{2}$

Multiply $- 1$ by $- 5$ .

$f (- x) = x^{6} + 5 x^{5} - 9 {(- x)}^{4} + 81 {(- x)}^{3} - 108 {(- x)}^{2}$

Apply the product rule to $- x$ .

$f (- x) = x^{6} + 5 x^{5} - 9 ({(- 1)}^{4} x^{4}) + 81 {(- x)}^{3} - 108 {(- x)}^{2}$

Raise $- 1$ to the power of $4$ .

$f (- x) = x^{6} + 5 x^{5} - 9 (1 x^{4}) + 81 {(- x)}^{3} - 108 {(- x)}^{2}$

Multiply $x^{4}$ by $1$ .

$f (- x) = x^{6} + 5 x^{5} - 9 x^{4} + 81 {(- x)}^{3} - 108 {(- x)}^{2}$

Apply the product rule to $- x$ .

$f (- x) = x^{6} + 5 x^{5} - 9 x^{4} + 81 ({(- 1)}^{3} x^{3}) - 108 {(- x)}^{2}$

Raise $- 1$ to the power of $3$ .

$f (- x) = x^{6} + 5 x^{5} - 9 x^{4} + 81 (- x^{3}) - 108 {(- x)}^{2}$

Multiply $- 1$ by $81$ .

$f (- x) = x^{6} + 5 x^{5} - 9 x^{4} - 81 x^{3} - 108 {(- x)}^{2}$

Apply the product rule to $- x$ .

$f (- x) = x^{6} + 5 x^{5} - 9 x^{4} - 81 x^{3} - 108 ({(- 1)}^{2} x^{2})$

Raise $- 1$ to the power of $2$ .

$f (- x) = x^{6} + 5 x^{5} - 9 x^{4} - 81 x^{3} - 108 (1 x^{2})$

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

$f (- x) = x^{6} + 5 x^{5} - 9 x^{4} - 81 x^{3} - 108 x^{2}$

Step 4

A function is even if $f (- x) = f (x)$ .

Check if $f (- x) = f (x)$ .

Since $x^{6} + 5 x^{5} - 9 x^{4} - 81 x^{3} - 108 x^{2}$ $\neq$ $x^{6} - 5 x^{5} - 9 x^{4} + 81 x^{3} - 108 x^{2}$ , the function is not even.

The function is not even

Step 5

A function is odd if $f (- x) = - f (x)$ .

Find $- f (x)$ .

Multiply $x^{6} - 5 x^{5} - 9 x^{4} + 81 x^{3} - 108 x^{2}$ by $- 1$ .

$- f (x) = - (x^{6} - 5 x^{5} - 9 x^{4} + 81 x^{3} - 108 x^{2})$

Apply the distributive property.

$- f (x) = - x^{6} - (- 5 x^{5}) - (- 9 x^{4}) - (81 x^{3}) - (- 108 x^{2})$

Simplify.

Multiply $- 5$ by $- 1$ .

$- f (x) = - x^{6} + 5 x^{5} - (- 9 x^{4}) - (81 x^{3}) - (- 108 x^{2})$

Multiply $- 9$ by $- 1$ .

$- f (x) = - x^{6} + 5 x^{5} + 9 x^{4} - (81 x^{3}) - (- 108 x^{2})$

Multiply $81$ by $- 1$ .

$- f (x) = - x^{6} + 5 x^{5} + 9 x^{4} - 81 x^{3} - (- 108 x^{2})$

Multiply $- 108$ by $- 1$ .

$- f (x) = - x^{6} + 5 x^{5} + 9 x^{4} - 81 x^{3} + 108 x^{2}$

Since $x^{6} + 5 x^{5} - 9 x^{4} - 81 x^{3} - 108 x^{2}$ $\neq$ $- x^{6} + 5 x^{5} + 9 x^{4} - 81 x^{3} + 108 x^{2}$ , the function is not odd.

The function is not odd

Step 6

The function is neither odd nor even

Step 7

Since the function is not odd, it is not symmetric about the origin.

No origin symmetry

Step 8

Since the function is not even, it is not symmetric about the y-axis.

No y-axis symmetry

Step 9

Since the function is neither odd nor even, there is no origin / y-axis symmetry.

Function is not symmetric

Step 10

Find the Symmetry f(x)=x^2(x-3)^3(x+4)

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