Algebra Examples

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

Step 1

Simplify and reorder the polynomial.

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Step 1.1

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

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

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ .

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

Step 1.3.1.2

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

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

Step 1.3.1.3

Multiply $- 3$ by $- 3$ .

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

Step 1.3.2

Subtract $3 x$ from $- 3 x$ .

$4 (x^{2} - 6 x + 9) (x^{2} + 1)$

Step 1.4

Apply the distributive property.

$(4 x^{2} + 4 (- 6 x) + 4 \cdot 9) (x^{2} + 1)$

Step 1.5

Simplify.

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Step 1.5.1

Multiply $- 6$ by $4$ .

$(4 x^{2} - 24 x + 4 \cdot 9) (x^{2} + 1)$

Step 1.5.2

Multiply $4$ by $9$ .

$(4 x^{2} - 24 x + 36) (x^{2} + 1)$

Step 1.6

Expand $(4 x^{2} - 24 x + 36) (x^{2} + 1)$ by multiplying each term in the first expression by each term in the second expression.

$4 x^{2} x^{2} + 4 x^{2} \cdot 1 - 24 x \cdot x^{2} - 24 x \cdot 1 + 36 x^{2} + 36 \cdot 1$

Step 1.7

Simplify terms.

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Step 1.7.1

Simplify each term.

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Step 1.7.1.1

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

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Step 1.7.1.1.1

Move $x^{2}$ .

$4 (x^{2} x^{2}) + 4 x^{2} \cdot 1 - 24 x \cdot x^{2} - 24 x \cdot 1 + 36 x^{2} + 36 \cdot 1$

Step 1.7.1.1.2

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

$4 x^{2 + 2} + 4 x^{2} \cdot 1 - 24 x \cdot x^{2} - 24 x \cdot 1 + 36 x^{2} + 36 \cdot 1$

Step 1.7.1.1.3

Add $2$ and $2$ .

$4 x^{4} + 4 x^{2} \cdot 1 - 24 x \cdot x^{2} - 24 x \cdot 1 + 36 x^{2} + 36 \cdot 1$

Step 1.7.1.2

Multiply $4$ by $1$ .

$4 x^{4} + 4 x^{2} - 24 x \cdot x^{2} - 24 x \cdot 1 + 36 x^{2} + 36 \cdot 1$

Step 1.7.1.3

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

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Step 1.7.1.3.1

Move $x^{2}$ .

$4 x^{4} + 4 x^{2} - 24 (x^{2} x) - 24 x \cdot 1 + 36 x^{2} + 36 \cdot 1$

Step 1.7.1.3.2

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

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Step 1.7.1.3.2.1

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

$4 x^{4} + 4 x^{2} - 24 (x^{2} x^{1}) - 24 x \cdot 1 + 36 x^{2} + 36 \cdot 1$

Step 1.7.1.3.2.2

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

$4 x^{4} + 4 x^{2} - 24 x^{2 + 1} - 24 x \cdot 1 + 36 x^{2} + 36 \cdot 1$

Step 1.7.1.3.3

Add $2$ and $1$ .

$4 x^{4} + 4 x^{2} - 24 x^{3} - 24 x \cdot 1 + 36 x^{2} + 36 \cdot 1$

Step 1.7.1.4

Multiply $- 24$ by $1$ .

$4 x^{4} + 4 x^{2} - 24 x^{3} - 24 x + 36 x^{2} + 36 \cdot 1$

Step 1.7.1.5

Multiply $36$ by $1$ .

$4 x^{4} + 4 x^{2} - 24 x^{3} - 24 x + 36 x^{2} + 36$

Step 1.7.2

Add $4 x^{2}$ and $36 x^{2}$ .

$4 x^{4} + 40 x^{2} - 24 x^{3} - 24 x + 36$

Step 2

The largest exponent is the degree of the polynomial.

$4$