Pre-Algebra Examples

$\frac{t^{2} + 4 t + 3}{2 t^{2} - t - 10} \cdot \frac{t}{t^{2} + 3 t + 2} \cdot \frac{2 t^{2} + 4 t^{3}}{t^{2} + 3 t}$

Step 1

Simplify with factoring out.

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Step 1.1

Factor $2 t^{2}$ out of $2 t^{2} + 4 t^{3}$ .

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Step 1.1.1

Factor $2 t^{2}$ out of $2 t^{2}$ .

$\frac{t^{2} + 4 t + 3}{2 t^{2} - t - 10} \cdot \frac{t}{t^{2} + 3 t + 2} \cdot \frac{2 t^{2} (1) + 4 t^{3}}{t^{2} + 3 t}$

Step 1.1.2

Factor $2 t^{2}$ out of $4 t^{3}$ .

$\frac{t^{2} + 4 t + 3}{2 t^{2} - t - 10} \cdot \frac{t}{t^{2} + 3 t + 2} \cdot \frac{2 t^{2} (1) + 2 t^{2} (2 t)}{t^{2} + 3 t}$

Step 1.1.3

Factor $2 t^{2}$ out of $2 t^{2} (1) + 2 t^{2} (2 t)$ .

$\frac{t^{2} + 4 t + 3}{2 t^{2} - t - 10} \cdot \frac{t}{t^{2} + 3 t + 2} \cdot \frac{2 t^{2} (1 + 2 t)}{t^{2} + 3 t}$

Step 1.2

Factor $t$ out of $t^{2} + 3 t$ .

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Step 1.2.1

Factor $t$ out of $t^{2}$ .

$\frac{t^{2} + 4 t + 3}{2 t^{2} - t - 10} \cdot \frac{t}{t^{2} + 3 t + 2} \cdot \frac{2 t^{2} (1 + 2 t)}{t \cdot t + 3 t}$

Step 1.2.2

Factor $t$ out of $3 t$ .

$\frac{t^{2} + 4 t + 3}{2 t^{2} - t - 10} \cdot \frac{t}{t^{2} + 3 t + 2} \cdot \frac{2 t^{2} (1 + 2 t)}{t \cdot t + t \cdot 3}$

Step 1.2.3

Factor $t$ out of $t \cdot t + t \cdot 3$ .

$\frac{t^{2} + 4 t + 3}{2 t^{2} - t - 10} \cdot \frac{t}{t^{2} + 3 t + 2} \cdot \frac{2 t^{2} (1 + 2 t)}{t (t + 3)}$

Step 2

Factor $t^{2} + 4 t + 3$ using the AC method.

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Step 2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $3$ and whose sum is $4$ .

$1, 3$

Step 2.2

Write the factored form using these integers.

$\frac{(t + 1) (t + 3)}{2 t^{2} - t - 10} \cdot \frac{t}{t^{2} + 3 t + 2} \cdot \frac{2 t^{2} (1 + 2 t)}{t (t + 3)}$

Step 3

Factor by grouping.

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Step 3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 10 = - 20$ and whose sum is $b = - 1$ .

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Step 3.1.1

Factor $- 1$ out of $- t$ .

$\frac{(t + 1) (t + 3)}{2 t^{2} - (t) - 10} \cdot \frac{t}{t^{2} + 3 t + 2} \cdot \frac{2 t^{2} (1 + 2 t)}{t (t + 3)}$

Step 3.1.2

Rewrite $- 1$ as $4$ plus $- 5$

$\frac{(t + 1) (t + 3)}{2 t^{2} + (4 - 5) t - 10} \cdot \frac{t}{t^{2} + 3 t + 2} \cdot \frac{2 t^{2} (1 + 2 t)}{t (t + 3)}$

Step 3.1.3

Apply the distributive property.

$\frac{(t + 1) (t + 3)}{2 t^{2} + 4 t - 5 t - 10} \cdot \frac{t}{t^{2} + 3 t + 2} \cdot \frac{2 t^{2} (1 + 2 t)}{t (t + 3)}$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$\frac{(t + 1) (t + 3)}{(2 t^{2} + 4 t) - 5 t - 10} \cdot \frac{t}{t^{2} + 3 t + 2} \cdot \frac{2 t^{2} (1 + 2 t)}{t (t + 3)}$

Step 3.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{(t + 1) (t + 3)}{2 t (t + 2) - 5 (t + 2)} \cdot \frac{t}{t^{2} + 3 t + 2} \cdot \frac{2 t^{2} (1 + 2 t)}{t (t + 3)}$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $t + 2$ .

$\frac{(t + 1) (t + 3)}{(t + 2) (2 t - 5)} \cdot \frac{t}{t^{2} + 3 t + 2} \cdot \frac{2 t^{2} (1 + 2 t)}{t (t + 3)}$

Step 4

Factor $t^{2} + 3 t + 2$ using the AC method.

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Step 4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $3$ .

$1, 2$

Step 4.2

Write the factored form using these integers.

$\frac{(t + 1) (t + 3)}{(t + 2) (2 t - 5)} \cdot \frac{t}{(t + 1) (t + 2)} \cdot \frac{2 t^{2} (1 + 2 t)}{t (t + 3)}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $t + 1$ .

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Step 5.1.1

Cancel the common factor.

$\frac{(t + 1) (t + 3)}{(t + 2) (2 t - 5)} \cdot \frac{t}{(t + 1) (t + 2)} \cdot \frac{2 t^{2} (1 + 2 t)}{t (t + 3)}$

Step 5.1.2

Rewrite the expression.

$\frac{t + 3}{(t + 2) (2 t - 5)} \cdot \frac{t}{t + 2} \cdot \frac{2 t^{2} (1 + 2 t)}{t (t + 3)}$

Step 5.2

Multiply $\frac{t + 3}{(t + 2) (2 t - 5)}$ by $\frac{t}{t + 2}$ .

$\frac{(t + 3) t}{(t + 2) (2 t - 5) (t + 2)} \cdot \frac{2 t^{2} (1 + 2 t)}{t (t + 3)}$

Step 6

Raise $t + 2$ to the power of $1$ .

$\frac{(t + 3) t}{{(t + 2)}^{1} (t + 2) (2 t - 5)} \cdot \frac{2 t^{2} (1 + 2 t)}{t (t + 3)}$

Step 7

Raise $t + 2$ to the power of $1$ .

$\frac{(t + 3) t}{{(t + 2)}^{1} {(t + 2)}^{1} (2 t - 5)} \cdot \frac{2 t^{2} (1 + 2 t)}{t (t + 3)}$

Step 8

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

$\frac{(t + 3) t}{{(t + 2)}^{1 + 1} (2 t - 5)} \cdot \frac{2 t^{2} (1 + 2 t)}{t (t + 3)}$

Step 9

Add $1$ and $1$ .

$\frac{(t + 3) t}{{(t + 2)}^{2} (2 t - 5)} \cdot \frac{2 t^{2} (1 + 2 t)}{t (t + 3)}$

Step 10

Combine.

$\frac{(t + 3) t (2 t^{2} (1 + 2 t))}{{(t + 2)}^{2} (2 t - 5) (t (t + 3))}$

Step 11

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

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Step 11.1

Move $t^{2}$ .

$\frac{(t + 3) (t^{2} t) (2 (1 + 2 t))}{{(t + 2)}^{2} (2 t - 5) (t (t + 3))}$

Step 11.2

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

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Step 11.2.1

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

$\frac{(t + 3) (t^{2} t^{1}) (2 (1 + 2 t))}{{(t + 2)}^{2} (2 t - 5) (t (t + 3))}$

Step 11.2.2

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

$\frac{(t + 3) t^{2 + 1} (2 (1 + 2 t))}{{(t + 2)}^{2} (2 t - 5) (t (t + 3))}$

Step 11.3

Add $2$ and $1$ .

$\frac{(t + 3) t^{3} (2 (1 + 2 t))}{{(t + 2)}^{2} (2 t - 5) (t (t + 3))}$

Step 12

Reduce the expression by cancelling the common factors.

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Step 12.1

Cancel the common factor of $t + 3$ .

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Step 12.1.1

Cancel the common factor.

$\frac{(t + 3) t^{3} (2 (1 + 2 t))}{{(t + 2)}^{2} (2 t - 5) (t (t + 3))}$

Step 12.1.2

Rewrite the expression.

$\frac{t^{3} (2 (1 + 2 t))}{{(t + 2)}^{2} (2 t - 5) (t)}$

Step 12.2

Cancel the common factor of $t^{3}$ and $t$ .

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Step 12.2.1

Factor $t$ out of $t^{3} (2 (1 + 2 t))$ .

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

Step 12.2.2

Cancel the common factors.

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Step 12.2.2.1

Factor $t$ out of ${(t + 2)}^{2} (2 t - 5) (t)$ .

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

Step 12.2.2.2

Cancel the common factor.

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

Step 12.2.2.3

Rewrite the expression.

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

Step 13

Simplify the numerator.

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Step 13.1

Rewrite.

$\frac{(t + 3) t (2 t^{2} (1 + 2 t))}{{(t + 2)}^{2} (2 t - 5)}$

Step 13.2

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

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Step 13.2.1

Move $t^{2}$ .

$\frac{(t + 3) (t^{2} t) (2 (1 + 2 t))}{{(t + 2)}^{2} (2 t - 5)}$

Step 13.2.2

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

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Step 13.2.2.1

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

$\frac{(t + 3) (t^{2} t^{1}) (2 (1 + 2 t))}{{(t + 2)}^{2} (2 t - 5)}$

Step 13.2.2.2

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

$\frac{(t + 3) t^{2 + 1} (2 (1 + 2 t))}{{(t + 2)}^{2} (2 t - 5)}$

Step 13.2.3

Add $2$ and $1$ .

$\frac{(t + 3) t^{3} (2 (1 + 2 t))}{{(t + 2)}^{2} (2 t - 5)}$

Step 13.3

Rewrite.

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

Step 13.4

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

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Step 13.4.1

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

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

Step 13.4.2

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

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

Step 13.5

Add $2$ and $1$ .

$\frac{t^{3} (2 (1 + 2 t))}{{(t + 2)}^{2} (2 t - 5)}$

Step 13.6

Factor $t$ out of $t^{3}$ .

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

Step 13.7

Rewrite.

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

Step 13.8

Remove unnecessary parentheses.

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

Step 14

Move $2$ to the left of $t^{2}$ .

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