Pre-Algebra Examples

$- 4 t^{2} + 19 t - 21$

Step 1

Factor $- 1$ out of $- 4 t^{2} + 19 t - 21$ .

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Step 1.1

Factor $- 1$ out of $- 4 t^{2}$ .

$- (4 t^{2}) + 19 t - 21$

Step 1.2

Factor $- 1$ out of $19 t$ .

$- (4 t^{2}) - (- 19 t) - 21$

Step 1.3

Rewrite $- 21$ as $- 1 (21)$ .

$- (4 t^{2}) - (- 19 t) - 1 (21)$

Step 1.4

Factor $- 1$ out of $- (4 t^{2}) - (- 19 t)$ .

$- (4 t^{2} - 19 t) - 1 (21)$

Step 1.5

Factor $- 1$ out of $- (4 t^{2} - 19 t) - 1 (21)$ .

$- (4 t^{2} - 19 t + 21)$

Step 2

Factor.

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Step 2.1

Factor by grouping.

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Step 2.1.1

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 $a \cdot c = 4 \cdot 21 = 84$ and whose sum is $b = - 19$ .

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Step 2.1.1.1

Factor $- 19$ out of $- 19 t$ .

$- (4 t^{2} - 19 (t) + 21)$

Step 2.1.1.2

Rewrite $- 19$ as $- 7$ plus $- 12$

$- (4 t^{2} + (- 7 - 12) t + 21)$

Step 2.1.1.3

Apply the distributive property.

$- (4 t^{2} - 7 t - 12 t + 21)$

Step 2.1.2

Factor out the greatest common factor from each group.

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Step 2.1.2.1

Group the first two terms and the last two terms.

$- ((4 t^{2} - 7 t) - 12 t + 21)$

Step 2.1.2.2

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

$- (t (4 t - 7) - 3 (4 t - 7))$

Step 2.1.3

Factor the polynomial by factoring out the greatest common factor, $4 t - 7$ .

$- ((4 t - 7) (t - 3))$

Step 2.2

Remove unnecessary parentheses.

$- (4 t - 7) (t - 3)$