Algebra Examples

$(4 a^{2} h^{2} - 8 a^{3} h + 3 a^{4}) \div (2 a^{2})$

Step 1

Rewrite the division as a fraction.

$\frac{4 a^{2} h^{2} - 8 a^{3} h + 3 a^{4}}{2 a^{2}}$

Step 2

Simplify the numerator.

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Step 2.1

Factor $a^{2}$ out of $4 a^{2} h^{2} - 8 a^{3} h + 3 a^{4}$ .

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Step 2.1.1

Factor $a^{2}$ out of $4 a^{2} h^{2}$ .

$\frac{a^{2} (4 h^{2}) - 8 a^{3} h + 3 a^{4}}{2 a^{2}}$

Step 2.1.2

Factor $a^{2}$ out of $- 8 a^{3} h$ .

$\frac{a^{2} (4 h^{2}) + a^{2} (- 8 a h) + 3 a^{4}}{2 a^{2}}$

Step 2.1.3

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

$\frac{a^{2} (4 h^{2}) + a^{2} (- 8 a h) + a^{2} (3 a^{2})}{2 a^{2}}$

Step 2.1.4

Factor $a^{2}$ out of $a^{2} (4 h^{2}) + a^{2} (- 8 a h)$ .

$\frac{a^{2} (4 h^{2} - 8 a h) + a^{2} (3 a^{2})}{2 a^{2}}$

Step 2.1.5

Factor $a^{2}$ out of $a^{2} (4 h^{2} - 8 a h) + a^{2} (3 a^{2})$ .

$\frac{a^{2} (4 h^{2} - 8 a h + 3 a^{2})}{2 a^{2}}$

Step 2.2

Factor by grouping.

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Step 2.2.1

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 $a \cdot c = 3 \cdot 4 = 12$ and whose sum is $b = - 8$ .

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Step 2.2.1.1

Reorder terms.

$\frac{a^{2} (3 a^{2} + 4 h^{2} - 8 a h)}{2 a^{2}}$

Step 2.2.1.2

Reorder $4 h^{2}$ and $- 8 a h$ .

$\frac{a^{2} (3 a^{2} - 8 a h + 4 h^{2})}{2 a^{2}}$

Step 2.2.1.3

Factor $- 8$ out of $- 8 a h$ .

$\frac{a^{2} (3 a^{2} - 8 (a h) + 4 h^{2})}{2 a^{2}}$

Step 2.2.1.4

Rewrite $- 8$ as $- 2$ plus $- 6$

$\frac{a^{2} (3 a^{2} + (- 2 - 6) (a h) + 4 h^{2})}{2 a^{2}}$

Step 2.2.1.5

Apply the distributive property.

$\frac{a^{2} (3 a^{2} - 2 (a h) - 6 (a h) + 4 h^{2})}{2 a^{2}}$

Step 2.2.1.6

Move parentheses.

$\frac{a^{2} (3 a^{2} - 2 a h - 6 a h + 4 h^{2})}{2 a^{2}}$

Step 2.2.2

Factor out the greatest common factor from each group.

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Step 2.2.2.1

Group the first two terms and the last two terms.

$\frac{a^{2} ((3 a^{2} - 2 a h) - 6 a h + 4 h^{2})}{2 a^{2}}$

Step 2.2.2.2

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

$\frac{a^{2} (a (3 a - 2 h) - 2 h (3 a - 2 h))}{2 a^{2}}$

Step 2.2.3

Factor the polynomial by factoring out the greatest common factor, $3 a - 2 h$ .

$\frac{a^{2} ((3 a - 2 h) (a - 2 h))}{2 a^{2}}$

$\frac{a^{2} (3 a - 2 h) (a - 2 h)}{2 a^{2}}$

Step 3

Cancel the common factor of $a^{2}$ .

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Step 3.1

Cancel the common factor.

$\frac{a^{2} (3 a - 2 h) (a - 2 h)}{2 a^{2}}$

Step 3.2

Rewrite the expression.

$\frac{(3 a - 2 h) (a - 2 h)}{2}$

Step 4

Expand $(3 a - 2 h) (a - 2 h)$ using the FOIL Method.

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Step 4.1

Apply the distributive property.

$\frac{3 a (a - 2 h) - 2 h (a - 2 h)}{2}$

Step 4.2

Apply the distributive property.

$\frac{3 a \cdot a + 3 a (- 2 h) - 2 h (a - 2 h)}{2}$

Step 4.3

Apply the distributive property.

$\frac{3 a \cdot a + 3 a (- 2 h) - 2 h a - 2 h (- 2 h)}{2}$

Step 5

Simplify and combine like terms.

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Step 5.1

Simplify each term.

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Step 5.1.1

Multiply $a$ by $a$ by adding the exponents.

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Step 5.1.1.1

Move $a$ .

$\frac{3 (a \cdot a) + 3 a (- 2 h) - 2 h a - 2 h (- 2 h)}{2}$

Step 5.1.1.2

Multiply $a$ by $a$ .

$\frac{3 a^{2} + 3 a (- 2 h) - 2 h a - 2 h (- 2 h)}{2}$

Step 5.1.2

Rewrite using the commutative property of multiplication.

$\frac{3 a^{2} + 3 \cdot - 2 a h - 2 h a - 2 h (- 2 h)}{2}$

Step 5.1.3

Multiply $3$ by $- 2$ .

$\frac{3 a^{2} - 6 a h - 2 h a - 2 h (- 2 h)}{2}$

Step 5.1.4

Rewrite using the commutative property of multiplication.

$\frac{3 a^{2} - 6 a h - 2 h a - 2 \cdot - 2 h \cdot h}{2}$

Step 5.1.5

Multiply $h$ by $h$ by adding the exponents.

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Step 5.1.5.1

Move $h$ .

$\frac{3 a^{2} - 6 a h - 2 h a - 2 \cdot - 2 (h \cdot h)}{2}$

Step 5.1.5.2

Multiply $h$ by $h$ .

$\frac{3 a^{2} - 6 a h - 2 h a - 2 \cdot - 2 h^{2}}{2}$

Step 5.1.6

Multiply $- 2$ by $- 2$ .

$\frac{3 a^{2} - 6 a h - 2 h a + 4 h^{2}}{2}$

Step 5.2

Subtract $2 h a$ from $- 6 a h$ .

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Step 5.2.1

Move $h$ .

$\frac{3 a^{2} - 6 a h - 2 a h + 4 h^{2}}{2}$

Step 5.2.2

Subtract $2 a h$ from $- 6 a h$ .

$\frac{3 a^{2} - 8 a h + 4 h^{2}}{2}$

Step 6

Split the fraction $\frac{3 a^{2} - 8 a h + 4 h^{2}}{2}$ into two fractions.

$\frac{3 a^{2} - 8 a h}{2} + \frac{4 h^{2}}{2}$

Step 7

Split the fraction $\frac{3 a^{2} - 8 a h}{2}$ into two fractions.

$\frac{3 a^{2}}{2} + \frac{- 8 a h}{2} + \frac{4 h^{2}}{2}$

Step 8

Cancel the common factor of $- 8$ and $2$ .

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Step 8.1

Factor $2$ out of $- 8 a h$ .

$\frac{3 a^{2}}{2} + \frac{2 (- 4 a h)}{2} + \frac{4 h^{2}}{2}$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $2$ out of $2$ .

$\frac{3 a^{2}}{2} + \frac{2 (- 4 a h)}{2 (1)} + \frac{4 h^{2}}{2}$

Step 8.2.2

Cancel the common factor.

$\frac{3 a^{2}}{2} + \frac{2 (- 4 a h)}{2 \cdot 1} + \frac{4 h^{2}}{2}$

Step 8.2.3

Rewrite the expression.

$\frac{3 a^{2}}{2} + \frac{- 4 a h}{1} + \frac{4 h^{2}}{2}$

Step 8.2.4

Divide $- 4 a h$ by $1$ .

$\frac{3 a^{2}}{2} - 4 a h + \frac{4 h^{2}}{2}$

Step 9

Cancel the common factor of $4$ and $2$ .

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Step 9.1

Factor $2$ out of $4 h^{2}$ .

$\frac{3 a^{2}}{2} - 4 a h + \frac{2 (2 h^{2})}{2}$

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $2$ out of $2$ .

$\frac{3 a^{2}}{2} - 4 a h + \frac{2 (2 h^{2})}{2 (1)}$

Step 9.2.2

Cancel the common factor.

$\frac{3 a^{2}}{2} - 4 a h + \frac{2 (2 h^{2})}{2 \cdot 1}$

Step 9.2.3

Rewrite the expression.

$\frac{3 a^{2}}{2} - 4 a h + \frac{2 h^{2}}{1}$

Step 9.2.4

Divide $2 h^{2}$ by $1$ .

$\frac{3 a^{2}}{2} - 4 a h + 2 h^{2}$