Algebra Examples

$x^{3} - 3 x^{2} - 10 x - x^{2} + 3 x + 10 = 0$

Step 1

Regroup terms.

$x^{3} - 3 x^{2} - 10 x - x^{2} + 3 x + 10 = 0$

Step 2

Factor $x$ out of $x^{3} - 3 x^{2} - 10 x$ .

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Step 2.1

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

$x \cdot x^{2} - 3 x^{2} - 10 x - x^{2} + 3 x + 10 = 0$

Step 2.2

Factor $x$ out of $- 3 x^{2}$ .

$x \cdot x^{2} + x (- 3 x) - 10 x - x^{2} + 3 x + 10 = 0$

Step 2.3

Factor $x$ out of $- 10 x$ .

$x \cdot x^{2} + x (- 3 x) + x \cdot - 10 - x^{2} + 3 x + 10 = 0$

Step 2.4

Factor $x$ out of $x \cdot x^{2} + x (- 3 x)$ .

$x (x^{2} - 3 x) + x \cdot - 10 - x^{2} + 3 x + 10 = 0$

Step 2.5

Factor $x$ out of $x (x^{2} - 3 x) + x \cdot - 10$ .

$x (x^{2} - 3 x - 10) - x^{2} + 3 x + 10 = 0$

Step 3

Factor.

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Step 3.1

Factor $x^{2} - 3 x - 10$ using the AC method.

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Step 3.1.1

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 $- 10$ and whose sum is $- 3$ .

$- 5, 2$

Step 3.1.2

Write the factored form using these integers.

$x ((x - 5) (x + 2)) - x^{2} + 3 x + 10 = 0$

Step 3.2

Remove unnecessary parentheses.

$x (x - 5) (x + 2) - x^{2} + 3 x + 10 = 0$

Step 4

Factor by grouping.

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Step 4.1

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 $a \cdot c = - 1 \cdot 10 = - 10$ and whose sum is $b = 3$ .

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Step 4.1.1

Factor $3$ out of $3 x$ .

$x (x - 5) (x + 2) - x^{2} + 3 (x) + 10 = 0$

Step 4.1.2

Rewrite $3$ as $- 2$ plus $5$

$x (x - 5) (x + 2) - x^{2} + (- 2 + 5) x + 10 = 0$

Step 4.1.3

Apply the distributive property.

$x (x - 5) (x + 2) - x^{2} - 2 x + 5 x + 10 = 0$

Step 4.2

Factor out the greatest common factor from each group.

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Step 4.2.1

Group the first two terms and the last two terms.

$x (x - 5) (x + 2) + (- x^{2} - 2 x) + 5 x + 10 = 0$

Step 4.2.2

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

$x (x - 5) (x + 2) + x (- x - 2) - 5 (- x - 2) = 0$

Step 4.3

Factor the polynomial by factoring out the greatest common factor, $- x - 2$ .

$x (x - 5) (x + 2) + (- x - 2) (x - 5) = 0$

Step 5

Factor $x - 5$ out of $x (x - 5) (x + 2) + (- x - 2) (x - 5)$ .

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Step 5.1

Factor $x - 5$ out of $x (x - 5) (x + 2)$ .

$(x - 5) (x (x + 2)) + (- x - 2) (x - 5) = 0$

Step 5.2

Factor $x - 5$ out of $(- x - 2) (x - 5)$ .

$(x - 5) (x (x + 2)) + (x - 5) (- x - 2) = 0$

Step 5.3

Factor $x - 5$ out of $(x - 5) (x (x + 2)) + (x - 5) (- x - 2)$ .

$(x - 5) (x (x + 2) - x - 2) = 0$

Step 6

Apply the distributive property.

$(x - 5) (x \cdot x + x \cdot 2 - x - 2) = 0$

Step 7

Multiply $x$ by $x$ .

$(x - 5) (x^{2} + x \cdot 2 - x - 2) = 0$

Step 8

Move $2$ to the left of $x$ .

$(x - 5) (x^{2} + 2 x - x - 2) = 0$

Step 9

Subtract $x$ from $2 x$ .

$(x - 5) (x^{2} + x - 2) = 0$

Step 10

Factor.

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Step 10.1

Factor $x^{2} + x - 2$ using the AC method.

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Step 10.1.1

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 $1$ .

$- 1, 2$

Step 10.1.2

Write the factored form using these integers.

$(x - 5) ((x - 1) (x + 2)) = 0$

Step 10.2

Remove unnecessary parentheses.

$(x - 5) (x - 1) (x + 2) = 0$