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Polynomial Factoring: Learn Techniques & Solve Equations

Understand factoring polynomials into prime factors, use zero product property to solve equations, master quadratic equations, and factor perfect squares. Examples and steps included.

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Polynomial Factoring: Learn Techniques & Solve Equations

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  1. Lesson 23 Factoring polynomials

  2. 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.

  3. Factoring out the Greatest Common Monomial factor • Factor: • 3x2 + 6x -3 • 3(x2+2x-1) • 6x2 + 18x + 24 • 6(x2 + 3x + 4) • -4x3-10x = -2 2 xxx- 2 5 x • -2x (2x2+ 5)

  4. Factoring polynomials • Type pattern example • Basic x2 + bx +c= x2 + 9x +20 • Quadratic(x+u)(x+v)(x+4)(x+5) • Trinomial u+v=b,uv=c 4+5=9,4x5=20 • Perfect a2 +2ab+b2= x2 +8x +16= • Square(a+b)(a+b) (x+4)(x+4) • Trinomial a2-2ab+b2= x2-8x+16= • (a-b)(a-b) (x-4)(x-4) • Difference a2-b2= x2-9= • Of two (a-b)(a+b) (x-3)(x+3) • squares

  5. Factoring a perfect square trinomial • Factor: • x2-10x +25 • 4m2+12mn+9n2 • x2 +14x+49 • 9x2 +30xy + 25y2

  6. Factoring a difference of 2 squares • Factor: • x2 -9 • x2 -25 • 4x2 - 49 • 25x2 - 36

  7. 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

  8. Solving quadratic equations • Solve: • x2 + 3x -10 =0 • Factor (x+5)(x-2) = 0 • So x+5 = 0 or x-2 = 0 • x = -5 or x = 2 • x2 +2x -24 =0 • Factor (x+6)(x-4) = 0 • So x+6 = 0 or x-4 = 0 • x=-6 or x = 4

  9. solve • x2 -9 = 0 • 9x2 - 49 = 0 • x2 - 10x +21 = 0 • x2 - 1 = 0 • x2 +12x + 36 =0

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