Find the Degree f(x)=(x-6)^2(x+2)^2

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

Step 1

Simplify and reorder the polynomial.

Rewrite ${(x - 6)}^{2}$ as $(x - 6) (x - 6)$ .

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 6$ to the left of $x$ .

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

Multiply $- 6$ by $- 6$ .

$(x^{2} - 6 x - 6 x + 36) {(x + 2)}^{2}$

Subtract $6 x$ from $- 6 x$ .

$(x^{2} - 12 x + 36) {(x + 2)}^{2}$

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

$(x^{2} - 12 x + 36) ((x + 2) (x + 2))$

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

Apply the distributive property.

$(x^{2} - 12 x + 36) (x (x + 2) + 2 (x + 2))$

Apply the distributive property.

$(x^{2} - 12 x + 36) (x \cdot x + x \cdot 2 + 2 (x + 2))$

Apply the distributive property.

$(x^{2} - 12 x + 36) (x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$(x^{2} - 12 x + 36) (x^{2} + x \cdot 2 + 2 x + 2 \cdot 2)$

Move $2$ to the left of $x$ .

$(x^{2} - 12 x + 36) (x^{2} + 2 \cdot x + 2 x + 2 \cdot 2)$

Multiply $2$ by $2$ .

$(x^{2} - 12 x + 36) (x^{2} + 2 x + 2 x + 4)$

Add $2 x$ and $2 x$ .

$(x^{2} - 12 x + 36) (x^{2} + 4 x + 4)$

Expand $(x^{2} - 12 x + 36) (x^{2} + 4 x + 4)$ by multiplying each term in the first expression by each term in the second expression.

$x^{2} x^{2} + x^{2} (4 x) + x^{2} \cdot 4 - 12 x \cdot x^{2} - 12 x (4 x) - 12 x \cdot 4 + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

Simplify 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.

$x^{2 + 2} + x^{2} (4 x) + x^{2} \cdot 4 - 12 x \cdot x^{2} - 12 x (4 x) - 12 x \cdot 4 + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

Add $2$ and $2$ .

$x^{4} + x^{2} (4 x) + x^{2} \cdot 4 - 12 x \cdot x^{2} - 12 x (4 x) - 12 x \cdot 4 + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

Rewrite using the commutative property of multiplication.

$x^{4} + 4 x^{2} x + x^{2} \cdot 4 - 12 x \cdot x^{2} - 12 x (4 x) - 12 x \cdot 4 + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

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

Move $x$ .

$x^{4} + 4 (x \cdot x^{2}) + x^{2} \cdot 4 - 12 x \cdot x^{2} - 12 x (4 x) - 12 x \cdot 4 + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

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

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

$x^{4} + 4 (x^{1} x^{2}) + x^{2} \cdot 4 - 12 x \cdot x^{2} - 12 x (4 x) - 12 x \cdot 4 + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

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

$x^{4} + 4 x^{1 + 2} + x^{2} \cdot 4 - 12 x \cdot x^{2} - 12 x (4 x) - 12 x \cdot 4 + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

Add $1$ and $2$ .

$x^{4} + 4 x^{3} + x^{2} \cdot 4 - 12 x \cdot x^{2} - 12 x (4 x) - 12 x \cdot 4 + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

Move $4$ to the left of $x^{2}$ .

$x^{4} + 4 x^{3} + 4 \cdot x^{2} - 12 x \cdot x^{2} - 12 x (4 x) - 12 x \cdot 4 + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

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

Move $x^{2}$ .

$x^{4} + 4 x^{3} + 4 x^{2} - 12 (x^{2} x) - 12 x (4 x) - 12 x \cdot 4 + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

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

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

$x^{4} + 4 x^{3} + 4 x^{2} - 12 (x^{2} x^{1}) - 12 x (4 x) - 12 x \cdot 4 + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

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

$x^{4} + 4 x^{3} + 4 x^{2} - 12 x^{2 + 1} - 12 x (4 x) - 12 x \cdot 4 + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

Add $2$ and $1$ .

$x^{4} + 4 x^{3} + 4 x^{2} - 12 x^{3} - 12 x (4 x) - 12 x \cdot 4 + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

Rewrite using the commutative property of multiplication.

$x^{4} + 4 x^{3} + 4 x^{2} - 12 x^{3} - 12 \cdot 4 x \cdot x - 12 x \cdot 4 + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

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

Move $x$ .

$x^{4} + 4 x^{3} + 4 x^{2} - 12 x^{3} - 12 \cdot 4 (x \cdot x) - 12 x \cdot 4 + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

Multiply $x$ by $x$ .

$x^{4} + 4 x^{3} + 4 x^{2} - 12 x^{3} - 12 \cdot 4 x^{2} - 12 x \cdot 4 + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

Multiply $- 12$ by $4$ .

$x^{4} + 4 x^{3} + 4 x^{2} - 12 x^{3} - 48 x^{2} - 12 x \cdot 4 + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

Multiply $4$ by $- 12$ .

$x^{4} + 4 x^{3} + 4 x^{2} - 12 x^{3} - 48 x^{2} - 48 x + 36 x^{2} + 36 (4 x) + 36 \cdot 4$

Multiply $4$ by $36$ .

$x^{4} + 4 x^{3} + 4 x^{2} - 12 x^{3} - 48 x^{2} - 48 x + 36 x^{2} + 144 x + 36 \cdot 4$

Multiply $36$ by $4$ .

$x^{4} + 4 x^{3} + 4 x^{2} - 12 x^{3} - 48 x^{2} - 48 x + 36 x^{2} + 144 x + 144$

Simplify by adding terms.

Subtract $12 x^{3}$ from $4 x^{3}$ .

$x^{4} - 8 x^{3} + 4 x^{2} - 48 x^{2} - 48 x + 36 x^{2} + 144 x + 144$

Subtract $48 x^{2}$ from $4 x^{2}$ .

$x^{4} - 8 x^{3} - 44 x^{2} - 48 x + 36 x^{2} + 144 x + 144$

Add $- 44 x^{2}$ and $36 x^{2}$ .

$x^{4} - 8 x^{3} - 8 x^{2} - 48 x + 144 x + 144$

Add $- 48 x$ and $144 x$ .

$x^{4} - 8 x^{3} - 8 x^{2} + 96 x + 144$

Step 2

The largest exponent is the degree of the polynomial.

$4$

Find the Degree f(x)=(x-6)^2(x+2)^2

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