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“Bottoms Up”

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Factoring ax 2 + bx + c. “Bottoms Up”. Step 1: multiply the constant (c) term by the coefficient (a), of the leading term Constant is 6 Leading term is 6 Therefore 6 * 6 is 36. 6 x 2 + 13x + 6.

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6 x 2 13x 6

Step 1: multiply the constant (c) term by the coefficient (a), of the leading term

Constant is 6

Leading term is 6

Therefore 6 * 6 is 36

6x2 + 13x + 6
6x 2 13x 6

Step 2: Rewrite the equation by replacing the constant with the number in step 1, and remove the leading coefficient.

x2 + 13x + 36

6x2 + 13x + 6
slide4

Step 3: Factor the new equation from

step 2: x2 + 13x +36

(x + 4)(x + 9)

x 4 x 9

Step 4: Divide each constant term in the factored form by the original coefficient of the leading term.

Original coefficient of the leading term: 6

(x + 4)(x + 9)
slide7

Step 5: Now we use the “bottoms up” method- Move the denominator into the numerator to get:

slide9

Remember you may check some equations using graphing calculator- type in the equation in y = and find where it crosses the x-axis. Not all equations cross the x-axis. This particular one crosses at x = -2/3 and x = -3/2. Look at these solutions and the end result of step 4, and then look at step 5. Factoring leads to solutions, the zeros, the roots.

10 x 2 x 3

Step 1: multiply the constant (c) term by the coefficient (a), of the leading term

Constant is -3

Leading term is 10

Therefore -3 * 10 is -30

10x2 - x - 3
10 x 2 x 31

Step 2: Rewrite the equation by replacing the constant with the number in step 1, and remove the leading coefficient.

x2 - x - 30

10x2 - x - 3
x 2 x 30

Step 3: Factor the new equation from

step 2: x2 - x - 30

(x + 5)(x - 6)

x2 - x - 30
x 5 x 6

Step 4: Divide each constant term in the factored form by the original coefficient of the leading term.

Original coefficient of the leading term: 10

(x + 5)(x - 6)

(x + 5)(x - 6)

10 10

slide14

Then reduce the fractions:

(x + 1)(x - 3)

2 5

slide15

Step 5: Now we use the “bottoms up” method- Move the denominator into the numerator to get:

(2x + 1)(5x – 3)

(x + 1)(x - 3)

2 5

slide16

Check it- use FOIL:

10x2 -6x + 5x – 3

10x2 - x – 3

(2x + 1)(5x – 3)

you try

6x2 - 2x – 28

Step 1: multiply the constant (c) term by the coefficient (a), of the leading term

Constant is -28

Leading term is 6

Therefore 6 * -28 is -168

You Try!
6 x 2 2x 28

Step 2: Rewrite the equation by replacing the constant with the number in step 1, and remove the leading coefficient.

x2 -2x -168

6x2 - 2x – 28
slide19

Step 3: Factor the new equation

x2 -2x -168

( x -14 )(x +12 )

x 14 x 12

Step 4: Divide each constant term in the factored form by the original coefficient of the leading term.

Original coefficient of the leading term: 6

( x -14 )(x +12 )

6 6

( x -14 )(x +12 )
x 14 x 12 6 6

Then reduce the fractions:

( x-7 )(x +2 )

3 1

Step 5: Now we use the “bottoms up”

( x -14 )(x +12 ) 6 6
slide22

( 3x-7 )(x +2 )

Check your answer with factoring (your choice).

Conversation with a caution

6x2 - 2x – 28

Purpose of factoring

quick steps

Step 1: multiply (c) (a),

Step 2: Rewrite the equation by replacing the constant with the number in step 1, and remove the leading coefficient.

Step 3: Factor the new equation

Step 4: Divide each constant term in the factored form by the original coefficient of the leading term and then reduce the fractions:

Step 5: Now we use the “bottoms up”

Quick steps!
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