Algebra II - Polynomial Factorization

8 flashcards · Mathematics · by luckydog789 (harrison holbrook)

1. a2+2ab+b2a^2+2ab+b^2a2+2ab+b2

   a2+2ab+b2=(a+b)2a^2+2ab+b^2=(a+b)^2a2+2ab+b2=(a+b)2

2. a2−b2a^2-b^2a2−b2

   a2−b2=(a+b)(a−b)a^2-b^2=(a+b)(a-b)a2−b2=(a+b)(a−b)

3. Conjugates

   A conjugate of a binomial is the binomial with the sign of the second term inverted. For example, the conjugate of 5x+25x+\sqrt{2}5x+2 ​ is 5x−25x-\sqrt{2}5x−2 ​ . By multiplying the original binomial by the conjugate, the new binomial can be written in the form a2−b2a^2-b^2a2−b2 . This is useful in eliminating radicals and imaginary units, especially when rationalizing the denominator (or just making it ∈R\in \mathbb{R}∈R ), which can be achieved by multiplying by conjugateconjugate\frac{\text…

4. Completing the Square

   For ax2+bx+c=0ax^2+bx+c=0ax2+bx+c=0 Move ccc to the right Factor out aaa on the left Add (b2)2(\frac{b}{2})^2(2b​)2 to the inside of the reduced polynomial on the left and a(b2)2a(\frac{b}{2})^2a(2b​)2 to the right Factor the reduced polynomial on the left using x2+2nx+n2=(x+n)2x^2+2nx+n^2=(x+n)^2x2+2nx+n2=(x+n)2 or x2−2nx+n2=(x−n)2x^2-2nx+n^2=(x-n)^2x2−2nx+n2=(x−n)2 Rearrange so the equation equals 0

5. Factoring by Grouping

   For any polynomial (typically of even number of terms, often with terms of the same degree) Not common, but split terms to achieve GCF (Greatest Common Factor) grouping Separate terms into groups by similar GCF Factor out GCF in each group Hopefully, each factored out group should share a similar term. If so, use the distributive property to factor it out

6. Quadratic Formula

   x=−b±b2−4ac2ax=\frac{-b \pm \sqrt{b^2-4ac}}{2a}x=2a−b±b2−4ac ​​

7. Sum-Product method for when a=1a=1a=1

   For x2+bx+cx^2+bx+cx2+bx+c , Find all factors of ccc Check if two of the factors of ccc and add to equal bbb If so, factor as (x+f1)(x+f2)(x+f_1)(x+f_2)(x+f1​)(x+f2​) , where f1f_1f1​ and f2f_2f2​ are the above factors of ccc

8. AC method for when a≠1a \neq 1a=1

   for ax2+bx+cax^2+bx+cax2+bx+c , Find all factors of a⋅ca \cdot ca⋅c Check if two of the factors add to bbb If so, rewrite as ax2+f1x+f2x+cax^2+f_1x+f_2x+cax2+f1​x+f2​x+c , where f1f_1f1​ and f2f_2f2​ are the above factors Factor by grouping