Finite Math Examples

$\frac{1}{25} + \frac{1}{10} t^{2} + \frac{1}{16} t^{4}$

Step 1

Rewrite the middle term.

$\frac{1}{25} + 2 (\frac{1}{5}) (\frac{1}{4} t^{2}) + 0 + \frac{1}{16} t^{4}$

Step 2

Rearrange terms.

$\frac{1}{25} + 2 (\frac{1}{5}) (\frac{1}{4} t^{2}) + \frac{1}{16} t^{4} + 0$

Step 3

Factor first three terms by perfect square rule.

${(\frac{1}{5} + \frac{1}{4} t^{2})}^{2} + 0$

Step 4

Combine $\frac{1}{4}$ and $t^{2}$ .

${(\frac{1}{5} + \frac{t^{2}}{4})}^{2} + 0$

Step 5

Rewrite ${(\frac{1}{5} + \frac{t^{2}}{4})}^{2}$ as $(\frac{1}{5} + \frac{t^{2}}{4}) (\frac{1}{5} + \frac{t^{2}}{4})$ .

$(\frac{1}{5} + \frac{t^{2}}{4}) (\frac{1}{5} + \frac{t^{2}}{4}) + 0$

Step 6

Expand $(\frac{1}{5} + \frac{t^{2}}{4}) (\frac{1}{5} + \frac{t^{2}}{4})$ using the FOIL Method.

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Step 6.1

Apply the distributive property.

$\frac{1}{5} (\frac{1}{5} + \frac{t^{2}}{4}) + \frac{t^{2}}{4} (\frac{1}{5} + \frac{t^{2}}{4}) + 0$

Step 6.2

Apply the distributive property.

$\frac{1}{5} \cdot \frac{1}{5} + \frac{1}{5} \cdot \frac{t^{2}}{4} + \frac{t^{2}}{4} (\frac{1}{5} + \frac{t^{2}}{4}) + 0$

Step 6.3

Apply the distributive property.

$\frac{1}{5} \cdot \frac{1}{5} + \frac{1}{5} \cdot \frac{t^{2}}{4} + \frac{t^{2}}{4} \cdot \frac{1}{5} + \frac{t^{2}}{4} \cdot \frac{t^{2}}{4} + 0$

Step 7

Simplify and combine like terms.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply $\frac{1}{5} \cdot \frac{1}{5}$ .

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Step 7.1.1.1

Multiply $\frac{1}{5}$ by $\frac{1}{5}$ .

$\frac{1}{5 \cdot 5} + \frac{1}{5} \cdot \frac{t^{2}}{4} + \frac{t^{2}}{4} \cdot \frac{1}{5} + \frac{t^{2}}{4} \cdot \frac{t^{2}}{4} + 0$

Step 7.1.1.2

Multiply $5$ by $5$ .

$\frac{1}{25} + \frac{1}{5} \cdot \frac{t^{2}}{4} + \frac{t^{2}}{4} \cdot \frac{1}{5} + \frac{t^{2}}{4} \cdot \frac{t^{2}}{4} + 0$

Step 7.1.2

Combine.

$\frac{1}{25} + \frac{1 t^{2}}{5 \cdot 4} + \frac{t^{2}}{4} \cdot \frac{1}{5} + \frac{t^{2}}{4} \cdot \frac{t^{2}}{4} + 0$

Step 7.1.3

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

$\frac{1}{25} + \frac{t^{2}}{5 \cdot 4} + \frac{t^{2}}{4} \cdot \frac{1}{5} + \frac{t^{2}}{4} \cdot \frac{t^{2}}{4} + 0$

Step 7.1.4

Multiply $5$ by $4$ .

$\frac{1}{25} + \frac{t^{2}}{20} + \frac{t^{2}}{4} \cdot \frac{1}{5} + \frac{t^{2}}{4} \cdot \frac{t^{2}}{4} + 0$

Step 7.1.5

Combine.

$\frac{1}{25} + \frac{t^{2}}{20} + \frac{t^{2} \cdot 1}{4 \cdot 5} + \frac{t^{2}}{4} \cdot \frac{t^{2}}{4} + 0$

Step 7.1.6

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

$\frac{1}{25} + \frac{t^{2}}{20} + \frac{t^{2}}{4 \cdot 5} + \frac{t^{2}}{4} \cdot \frac{t^{2}}{4} + 0$

Step 7.1.7

Multiply $4$ by $5$ .

$\frac{1}{25} + \frac{t^{2}}{20} + \frac{t^{2}}{20} + \frac{t^{2}}{4} \cdot \frac{t^{2}}{4} + 0$

Step 7.1.8

Combine.

$\frac{1}{25} + \frac{t^{2}}{20} + \frac{t^{2}}{20} + \frac{t^{2} t^{2}}{4 \cdot 4} + 0$

Step 7.1.9

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

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Step 7.1.9.1

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

$\frac{1}{25} + \frac{t^{2}}{20} + \frac{t^{2}}{20} + \frac{t^{2 + 2}}{4 \cdot 4} + 0$

Step 7.1.9.2

Add $2$ and $2$ .

$\frac{1}{25} + \frac{t^{2}}{20} + \frac{t^{2}}{20} + \frac{t^{4}}{4 \cdot 4} + 0$

Step 7.1.10

Multiply $4$ by $4$ .

$\frac{1}{25} + \frac{t^{2}}{20} + \frac{t^{2}}{20} + \frac{t^{4}}{16} + 0$

Step 7.2

Add $\frac{t^{2}}{20}$ and $\frac{t^{2}}{20}$ .

$\frac{1}{25} + 2 \frac{t^{2}}{20} + \frac{t^{4}}{16} + 0$

Step 8

Cancel the common factor of $2$ .

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Step 8.1

Factor $2$ out of $20$ .

$\frac{1}{25} + 2 \frac{t^{2}}{2 (10)} + \frac{t^{4}}{16} + 0$

Step 8.2

Cancel the common factor.

$\frac{1}{25} + 2 \frac{t^{2}}{2 \cdot 10} + \frac{t^{4}}{16} + 0$

Step 8.3

Rewrite the expression.

$\frac{1}{25} + \frac{t^{2}}{10} + \frac{t^{4}}{16} + 0$

Step 9

Add $\frac{1}{25} + \frac{t^{2}}{10} + \frac{t^{4}}{16}$ and $0$ .

$\frac{1}{25} + \frac{t^{2}}{10} + \frac{t^{4}}{16}$

Step 10

Factor using the perfect square rule.

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Step 10.1

Rewrite $1$ as $1^{2}$ .

$\frac{1^{2}}{25} + \frac{t^{2}}{10} + \frac{t^{4}}{16}$

Step 10.2

Rewrite $25$ as $5^{2}$ .

$\frac{1^{2}}{5^{2}} + \frac{t^{2}}{10} + \frac{t^{4}}{16}$

Step 10.3

Rewrite $\frac{1^{2}}{5^{2}}$ as ${(\frac{1}{5})}^{2}$ .

${(\frac{1}{5})}^{2} + \frac{t^{2}}{10} + \frac{t^{4}}{16}$

Step 10.4

Rewrite $t^{4}$ as ${(t^{2})}^{2}$ .

${(\frac{1}{5})}^{2} + \frac{t^{2}}{10} + \frac{{(t^{2})}^{2}}{16}$

Step 10.5

Rewrite $16$ as $4^{2}$ .

${(\frac{1}{5})}^{2} + \frac{t^{2}}{10} + \frac{{(t^{2})}^{2}}{4^{2}}$

Step 10.6

Rewrite $\frac{{(t^{2})}^{2}}{4^{2}}$ as ${(\frac{t^{2}}{4})}^{2}$ .

${(\frac{1}{5})}^{2} + \frac{t^{2}}{10} + {(\frac{t^{2}}{4})}^{2}$

Step 10.7

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$\frac{t^{2}}{10} = 2 \cdot \frac{1}{5} \cdot \frac{t^{2}}{4}$

Step 10.8

Rewrite the polynomial.

${(\frac{1}{5})}^{2} + 2 \cdot \frac{1}{5} \cdot \frac{t^{2}}{4} + {(\frac{t^{2}}{4})}^{2}$

Step 10.9

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = \frac{1}{5}$ and $b = \frac{t^{2}}{4}$ .

${(\frac{1}{5} + \frac{t^{2}}{4})}^{2}$