Pre-Algebra Examples

4 x^{2} + 23 x y + 15 y^{2}

$4 x^{2} + 23 x y + 15 y^{2}$

Step 1

For a polynomial of the form

a x^{2} + b x + c

$a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is

a \cdot c = 4 \cdot 15 = 60

$a \cdot c = 4 \cdot 15 = 60$ and whose sum is

b = 23

$b = 23$ .

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Step 1.1

Reorder terms.

4 x^{2} + 15 y^{2} + 23 x y

$4 x^{2} + 15 y^{2} + 23 x y$

Step 1.2

Reorder

15 y^{2}

$15 y^{2}$ and

23 x y

$23 x y$ .

4 x^{2} + 23 x y + 15 y^{2}

$4 x^{2} + 23 x y + 15 y^{2}$

Step 1.3

Factor

23

$23$ out of

23 x y

$23 x y$ .

4 x^{2} + 23 (x y) + 15 y^{2}

$4 x^{2} + 23 (x y) + 15 y^{2}$

Step 1.4

Rewrite

23

$23$ as

3

$3$ plus

20

$20$

4 x^{2} + (3 + 20) (x y) + 15 y^{2}

$4 x^{2} + (3 + 20) (x y) + 15 y^{2}$

Step 1.5

Apply the distributive property.

4 x^{2} + 3 (x y) + 20 (x y) + 15 y^{2}

$4 x^{2} + 3 (x y) + 20 (x y) + 15 y^{2}$

Step 1.6

Remove unnecessary parentheses.

4 x^{2} + 3 x y + 20 (x y) + 15 y^{2}

$4 x^{2} + 3 x y + 20 (x y) + 15 y^{2}$

Step 1.7

Remove unnecessary parentheses.

4 x^{2} + 3 x y + 20 x y + 15 y^{2}

$4 x^{2} + 3 x y + 20 x y + 15 y^{2}$

4 x^{2} + 3 x y + 20 x y + 15 y^{2}

$4 x^{2} + 3 x y + 20 x y + 15 y^{2}$

Step 2

Factor out the greatest common factor from each group.

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Step 2.1

Group the first two terms and the last two terms.

(4 x^{2} + 3 x y) + 20 x y + 15 y^{2}

$(4 x^{2} + 3 x y) + 20 x y + 15 y^{2}$

Step 2.2

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

x (4 x + 3 y) + 5 y (4 x + 3 y)

$x (4 x + 3 y) + 5 y (4 x + 3 y)$

x (4 x + 3 y) + 5 y (4 x + 3 y)

$x (4 x + 3 y) + 5 y (4 x + 3 y)$

Step 3

Factor the polynomial by factoring out the greatest common factor,

4 x + 3 y

$4 x + 3 y$ .

(4 x + 3 y) (x + 5 y)

$(4 x + 3 y) (x + 5 y)$