Pre-Algebra Examples

$(- 16 z^{3} + 137 z^{2} - 72 z) \div (z - 8)$

Step 1

Rewrite the division as a fraction.

$\frac{- 16 z^{3} + 137 z^{2} - 72 z}{z - 8}$

Step 2

Simplify the numerator.

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Step 2.1

Factor $z$ out of $- 16 z^{3} + 137 z^{2} - 72 z$ .

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Step 2.1.1

Factor $z$ out of $- 16 z^{3}$ .

$\frac{z (- 16 z^{2}) + 137 z^{2} - 72 z}{z - 8}$

Step 2.1.2

Factor $z$ out of $137 z^{2}$ .

$\frac{z (- 16 z^{2}) + z (137 z) - 72 z}{z - 8}$

Step 2.1.3

Factor $z$ out of $- 72 z$ .

$\frac{z (- 16 z^{2}) + z (137 z) + z \cdot - 72}{z - 8}$

Step 2.1.4

Factor $z$ out of $z (- 16 z^{2}) + z (137 z)$ .

$\frac{z (- 16 z^{2} + 137 z) + z \cdot - 72}{z - 8}$

Step 2.1.5

Factor $z$ out of $z (- 16 z^{2} + 137 z) + z \cdot - 72$ .

$\frac{z (- 16 z^{2} + 137 z - 72)}{z - 8}$

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 $a \cdot c = - 16 \cdot - 72 = 1152$ and whose sum is $b = 137$ .

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Step 2.2.1.1

Factor $137$ out of $137 z$ .

$\frac{z (- 16 z^{2} + 137 (z) - 72)}{z - 8}$

Step 2.2.1.2

Rewrite $137$ as $9$ plus $128$

$\frac{z (- 16 z^{2} + (9 + 128) z - 72)}{z - 8}$

Step 2.2.1.3

Apply the distributive property.

$\frac{z (- 16 z^{2} + 9 z + 128 z - 72)}{z - 8}$

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.

$\frac{z ((- 16 z^{2} + 9 z) + 128 z - 72)}{z - 8}$

Step 2.2.2.2

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

$\frac{z (z (- 16 z + 9) - 8 (- 16 z + 9))}{z - 8}$

Step 2.2.3

Factor the polynomial by factoring out the greatest common factor, $- 16 z + 9$ .

$\frac{z ((- 16 z + 9) (z - 8))}{z - 8}$

$\frac{z (- 16 z + 9) (z - 8)}{z - 8}$

Step 3

Cancel the common factor of $z - 8$ .

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Step 3.1

Cancel the common factor.

$\frac{z (- 16 z + 9) (z - 8)}{z - 8}$

Step 3.2

Divide $z (- 16 z + 9)$ by $1$ .

$z (- 16 z + 9)$

Step 4

Apply the distributive property.

$z (- 16 z) + z \cdot 9$

Step 5

Rewrite using the commutative property of multiplication.

$- 16 z \cdot z + z \cdot 9$

Step 6

Move $9$ to the left of $z$ .

$- 16 z \cdot z + 9 \cdot z$

Step 7

Multiply $z$ by $z$ by adding the exponents.

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Step 7.1

Move $z$ .

$- 16 (z \cdot z) + 9 \cdot z$

Step 7.2

Multiply $z$ by $z$ .

$- 16 z^{2} + 9 \cdot z$

$- 16 z^{2} + 9 z$