Factor z^2(z-x)+4zx(z-x)-5x^2(z-x)

$z^{2} (z - x) + 4 z x (z - x) - 5 x^{2} (z - x)$

Step 1

Factor $z - x$ out of $z^{2} (z - x) + 4 z x (z - x) - 5 x^{2} (z - x)$ .

Factor $z - x$ out of $z^{2} (z - x)$ .

$(z - x) z^{2} + 4 z x (z - x) - 5 x^{2} (z - x)$

Factor $z - x$ out of $4 z x (z - x)$ .

$(z - x) z^{2} + (z - x) (4 z x) - 5 x^{2} (z - x)$

Factor $z - x$ out of $- 5 x^{2} (z - x)$ .

$(z - x) z^{2} + (z - x) (4 z x) + (z - x) (- 5 x^{2})$

Factor $z - x$ out of $(z - x) z^{2} + (z - x) (4 z x)$ .

$(z - x) (z^{2} + 4 z x) + (z - x) (- 5 x^{2})$

Factor $z - x$ out of $(z - x) (z^{2} + 4 z x) + (z - x) (- 5 x^{2})$ .

$(z - x) (z^{2} + 4 z x - 5 x^{2})$

Step 2

Factor.

Factor by grouping.

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 $a \cdot c = - 5 \cdot 1 = - 5$ and whose sum is $b = 4$ .

Reorder terms.

$(z - x) (- 5 x^{2} + z^{2} + 4 x z)$

Reorder $z^{2}$ and $4 x z$ .

$(z - x) (- 5 x^{2} + 4 x z + z^{2})$

Factor $4$ out of $4 x z$ .

$(z - x) (- 5 x^{2} + 4 (x z) + z^{2})$

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

$(z - x) (- 5 x^{2} + (- 1 + 5) (x z) + z^{2})$

Apply the distributive property.

$(z - x) (- 5 x^{2} - 1 (x z) + 5 (x z) + z^{2})$

Remove unnecessary parentheses.

$(z - x) (- 5 x^{2} - 1 x z + 5 (x z) + z^{2})$

Remove unnecessary parentheses.

$(z - x) (- 5 x^{2} - 1 x z + 5 x z + z^{2})$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(z - x) ((- 5 x^{2} - 1 x z) + 5 x z + z^{2})$

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

$(z - x) (x (- 5 x - z) - z (- 5 x - z))$

Factor the polynomial by factoring out the greatest common factor, $- 5 x - z$ .

$(z - x) ((- 5 x - z) (x - z))$

Remove unnecessary parentheses.

$(z - x) (- 5 x - z) (x - z)$

Step 3

Combine exponents.

Factor $- 1$ out of $z$ .

$(- 1 (- z) - x) (- 5 x - z) (x - z)$

Factor $- 1$ out of $- x$ .

$(- 1 (- z) - (x)) (- 5 x - z) (x - z)$

Factor $- 1$ out of $- 1 (- z) - (x)$ .

$- 1 (- z + x) (- 5 x - z) (x - z)$

Reorder terms.

$- 1 (x - z) (- 5 x - z) (x - z)$

Raise $x - z$ to the power of $1$ .

$- 1 ({(x - z)}^{1} (x - z)) (- 5 x - z)$

Raise $x - z$ to the power of $1$ .

$- 1 ({(x - z)}^{1} {(x - z)}^{1}) (- 5 x - z)$

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

$- 1 {(x - z)}^{1 + 1} (- 5 x - z)$

Add $1$ and $1$ .

$- 1 {(x - z)}^{2} (- 5 x - z)$

Step 4

Factor.

Factor $- 1$ out of $- 5 x - z$ .

Factor $- 1$ out of $- 5 x$ .

$- 1 {(x - z)}^{2} (- (5 x) - z)$

Factor $- 1$ out of $- z$ .

$- 1 {(x - z)}^{2} (- (5 x) - (z))$

Factor $- 1$ out of $- (5 x) - (z)$ .

$- 1 {(x - z)}^{2} (- (5 x + z))$

Remove unnecessary parentheses.

$- 1 {(x - z)}^{2} \cdot - 1 (5 x + z)$

Step 5

Combine exponents.

Factor out negative.

$- ({(x - z)}^{2} \cdot - 1 (5 x + z))$

Multiply $- 1$ by $- 1$ .

$1 ({(x - z)}^{2} (5 x + z))$

Multiply ${(x - z)}^{2}$ by $1$ .

${(x - z)}^{2} (5 x + z)$

Factor z^2(z-x)+4zx(z-x)-5x^2(z-x)

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