Lesson 23 Factoring polynomials To completely factor a

Lesson 23 Factoring polynomials

• To completely factor a number means to write it as a product of prime factors. • To completely factor a polynomial is to express it as a factor of prime polynomial factors. • A prime polynomial is a polynomial that cannot be factored.

Factoring out the Greatest Common Monomial factor • Factor: • 3 x 2 + 6 x -3 • 3(x 2+2 x-1) • 6 x 2 + 18 x + 24 • 6(x 2 + 3 x + 4) • -4 x 3 -10 x = -2 2 xxx- 2 5 x • -2 x (2 x 2+ 5)

Factoring polynomials • • • Type pattern Basic x 2 + bx +c= Quadratic (x+u)(x+v) Trinomial u+v=b, uv=c Perfect a 2 +2 ab+b 2= Square (a+b) Trinomial a 2 -2 ab+b 2= (a-b) Difference a 2 -b 2= Of two (a-b)(a+b) squares example x 2 + 9 x +20 (x+4)(x+5) 4+5=9, 4 x 5=20 x 2 +8 x +16= (x+4) x 2 -8 x+16= (x-4) x 2 -9= (x-3)(x+3)

Factoring a perfect square trinomial • Factor: • x 2 -10 x +25 • 4 m 2+12 mn+9 n 2 • x 2 +14 x+49 • 9 x 2 +30 xy + 25 y 2

Factoring a difference of 2 squares • Factor: • x 2 -9 • x 2 -25 • 4 x 2 - 49 • 25 x 2 - 36

Zero product property • If a product is zero, then at least one of the factors must be zero. • If a and b are real numbers, • If ab = 0, then a = 0 or b = 0

Solving quadratic equations • • • Solve: x 2 + 3 x -10 =0 Factor (x+5)(x-2) = 0 So x+5 = 0 or x-2 = 0 x = -5 or x = 2 • x 2 +2 x -24 =0 • Factor (x+6)(x-4) = 0 • So x+6 = 0 or x-4 = 0 • x=-6 or x = 4

solve • x 2 -9 = 0 • 9 x 2 - 49 = 0 • x 2 - 10 x +21 = 0 • x 2 - 1 = 0 • x 2 +12 x + 36 =0