Solving Polynomial Equations Sum and Difference of Cubes

Solving Polynomial Equations

Sum and Difference of Cubes Cubic expressions with sums or differences have unique factoring patterns. Formulas: Examples: he t a + b = (a + b)(a - ab + b ) -See ence? r e f f di 2 2 3 3 a - b = (a - b)(a + ab + b ) 3 3 3 x + 27 3 3 (x) + 3 3 3 2 2 2 3 x - 64 Given 3 (x) - 4 Expand… x is a, and 3 is b 2 (x) + 3 = (x + 3)(x - 3 x + 3 ) (Factored Form) 3 3 3 Given Expand… x is a, and 4 is b 2 2 (x) - 3 = (x - 3)(x + 3 ) (Factored Form)

Solving Polynomial Equations Once you got the expression in factored form, you can solve for the roots. Using a T- chart. Example: 2 3 3 Given b + -5 - 4 ac r 125 x + 1 X= igge b y a nt. r t ’s cie Let coeffi 3 (5 x) + (1) 3 2 (5 x + 1)((5 x) - 5 x + 1 ) 2 5 x + 1 25 x - 5 x + 1 x=-1/5 X= 2(a) Expand 5 +- 5 2 7 2 (Factored Form) X= -(-5) +- Plug a, b, and c into the quadratic formula if the quadratic expression cannot be factored. X= 5 + - 25 - 200 2 X= 5 +- -175 2 -5 - 4(25)(1) 2(1)

Factoring a Higher-Degree Polynomial Equation Make the process easier than the look of the expression. 4 . 2 x - 4 x - 12 - 2 -6 is -12; & 2 - 6= 4 2 a - 4 a - 12 = (a + 2)(a - 6) 4 2 2 2 x - 4 x - 12 = (x + 2)(x - 6) Given Let a represent x 2 (Factored Form) Substitute a for x 2

Solving a Higher-Degree Polynomial Equation Same expression, same steps, only different. We are solving for the roots this time. 4 2 Given x - 4 x - 12 2 Let a represent x a - 4 a - 12 2 (Factored Form) a - 4 a - 12 = (a + 2)(a - 6) 4 2 2 2 x - 4 x - 12 = (x + 2)(x - 6) x=+ 6 x=i 2, -i - 2 , x=+ Substitute a for x - -2 6 , and - 2 T-chart 6 2

Good luck in the Future!