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Algebra II

Factoring, expansion, quadratic equations, rules of factors and expansion

Polynomials exist in two different forms: factored form or expanded form.

Factored form Expanded form
(x+2)(x+3) x2+5x+6

Going from factored form to expanded form is called expansion, while going from expanded form to factored form is called factoring.

Expansion

Expansion is easier to do than factoring, most of the time. To expand a polynomial, simply multiply each term in one factor by every term in the other factor.

For example, to expand the polynomial: (x+1)(x+2), the x in (x+1) gets multiplied with the x and the 2 in (x+2). After this, the terms (products) are added together.

Then the 1 gets multiplied with the x and 2 in (x+2).

The middle step would look like: x*x + x*2 + 1*x + 1*2

In a simpler form, it would look like: x2 + 2x + x + 2

The two middle terms are of the same degree so they can be added. The final result would be: x2 + 3x + 2

Factoring

Factoring is the reverse of expansion. Suppose you wanted to factor the following polynomial: 6x2 + 15x + 6

First, multiply the coefficient of the first term with the constant in the last term. The result would be 36.

Next, list all the pairs of factors that multiply to produce 36:

1 * 36
2 * 18
3 * 12
4 * 9
6 * 6

See which pair of factors add to produce the coefficient of the middle term. Notice that 3 and 12 add to produce 15.

Rewrite the polynomial with four terms. The factors will become the coefficients of the middle terms: 6x2 + 3x + 12x + 6

Factor out the common factor from the first two terms: 3x( 2x + 1 ) + 12x + 6

Then factor out the common factor from the last two terms: 3x( 2x + 1 ) + 6( 2x + 1 )

Notice that ( 2x + 1 ) is a common factor in the result. Finally, factor out ( 2x + 1 ) to get ( 3x + 6 ) ( 2x + 1 )

Rules for Factoring

Pascal's Triangle

To get a coefficient add the two coefficients from above. Also, the coefficients can be determined by the combinations formula in probability.

1. ( a ± b )0 = 1
2. ( a ± b )1 = a ± b
3. ( a ± b )2 = a2 ± 2ab + b2
4. ( a ± b )3 = a3 ± 3a2b + 3ab2 ± b3
5. ( a ± b )4 = a4 ± 4a3b + 6a2b2 ± 4ab3 + b4
6. ( a ± b )5 = a5 ± 5a4b + 10a3b2 ± 10a2b3 + 5ab4 ± b5
7. ( a ± b )6 = a6 ± 6a5b + 15a4b2 ± 20a3b3 + 15a2b4 ± 6ab5 + b6
8. ( a ± b )7 = a7 ± 7a6b + 21a5b2 ± 35a4b3 + 35a3b4 ± 21a2b5 + 7ab6 ± b7
9. ( a ± b )8 = a8 ± 8a7b + 28a6b2 ± 56a5b3 + 70a4b4 ± 56a3b5 + 28a2b6 ± 8ab7 + b8
10. ( a ± b )9 = a9 ± 9a8b + 36a7b2 ± 84a6b3 + 126a5b4 ± 126a4b5 + 84a3b6 ± 36a2b7 + 9ab8 ± b9

Additional Rules for Factoring

1. a2 - b2 = (a-b)(a+b)
2. a2 + b2 = (a-bi)(a+bi), where i = √( -1)
3. a3 - b3 = (a-b)(a2 + ab + b2 )
4. a3 + b3 = (a+b)(a2 - ab + b2 )
5. a4 + b4 = (a2 + ab√2 + b2)(a2 - ab√2 + b2)
6. an - bn = (a-b)(a(n-1) + a(n-2)b + ... + b(n-1) ), n is even
7. an - bn = (a+b)(a(n-1) - a(n-2)b + ... - b(n-1) ), n is odd
8. an + bn = (a+b)(a(n-1) - a(n-2)b + ... + b(n-1) )

Quadratic Equations

General or Standard Form

ax2 + bx + c = 0

Root Form

x  =   -b ± √( b2 - 4ac )
2a
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