Algebra Examples

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

Step 1

Rewrite ${(3 x^{4} + 1)}^{2}$ as $(3 x^{4} + 1) (3 x^{4} + 1)$ .

$f (x) = (3 x^{4} + 1) (3 x^{4} + 1)$

Step 2

Expand $(3 x^{4} + 1) (3 x^{4} + 1)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$f (x) = 3 x^{4} (3 x^{4} + 1) + 1 (3 x^{4} + 1)$

Step 2.2

Apply the distributive property.

$f (x) = 3 x^{4} (3 x^{4}) + 3 x^{4} \cdot 1 + 1 (3 x^{4} + 1)$

Step 2.3

Apply the distributive property.

$f (x) = 3 x^{4} (3 x^{4}) + 3 x^{4} \cdot 1 + 1 (3 x^{4}) + 1 \cdot 1$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite using the commutative property of multiplication.

$f (x) = 3 \cdot (3 x^{4} x^{4}) + 3 x^{4} \cdot 1 + 1 (3 x^{4}) + 1 \cdot 1$

Step 3.1.2

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

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Step 3.1.2.1

Move $x^{4}$ .

$f (x) = 3 \cdot (3 (x^{4} x^{4})) + 3 x^{4} \cdot 1 + 1 (3 x^{4}) + 1 \cdot 1$

Step 3.1.2.2

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

$f (x) = 3 \cdot (3 x^{4 + 4}) + 3 x^{4} \cdot 1 + 1 (3 x^{4}) + 1 \cdot 1$

Step 3.1.2.3

Add $4$ and $4$ .

$f (x) = 3 \cdot (3 x^{8}) + 3 x^{4} \cdot 1 + 1 (3 x^{4}) + 1 \cdot 1$

Step 3.1.3

Multiply $3$ by $3$ .

$f (x) = 9 x^{8} + 3 x^{4} \cdot 1 + 1 (3 x^{4}) + 1 \cdot 1$

Step 3.1.4

Multiply $3$ by $1$ .

$f (x) = 9 x^{8} + 3 x^{4} + 1 (3 x^{4}) + 1 \cdot 1$

Step 3.1.5

Multiply $3 x^{4}$ by $1$ .

$f (x) = 9 x^{8} + 3 x^{4} + 3 x^{4} + 1 \cdot 1$

Step 3.1.6

Multiply $1$ by $1$ .

$f (x) = 9 x^{8} + 3 x^{4} + 3 x^{4} + 1$

Step 3.2

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

$f (x) = 9 x^{8} + 6 x^{4} + 1$

Step 4

Identify the exponents on the variables in each term, and add them together to find the degree of each term.

$9 x^{8} \to 8$

$6 x^{4} \to 4$

$1 \to 0$

Step 5

The largest exponent is the degree of the polynomial.

$8$

Step 6

The maximum number of possible roots is the degree of $9 x^{8} + 6 x^{4} + 1$ .

$8$