5.1 Factoring – the Greatest Common Factor Finding the Greatest Common Factor: Factor – write each number in factored form. List common factors Choose the smallest exponents – for variables and prime factors Multiply the primes and variables from step 3 Always factor out the GCF first when factoring an expression

5.1 Factoring – the Greatest Common Factor Example: factor 5x2y + 25xy2z

5.1 Factoring – Factor By Grouping Factoring by grouping Group Terms – collect the terms in 2 groups that have a common factor Factor within groups Factor the entire polynomial – factor out a common binomial factor from step 2 If necessary rearrange terms – if step 3 didn’t work, repeat steps 2 & 3 until you get 2 binomial factors

5.1 Factoring – Factor By Grouping Example: This arrangement doesn’t work. Rearrange and try again

5.2 Factoring Trinomials Factoring x2 + bx + c (no “ax2” term yet) Find 2 integers: product is c and sum is b Both integers are positive if b and c are positive Both integers are negative if c is positive and b is negative One integer is positive and one is negative if c is negative

5.2 Factoring Trinomials Example:

5.3 Factoring Trinomials – Factor By Grouping Factoring ax2 + bx + c by grouping Multiply a times c Find a factorization of the number from step 1 that also adds up to b Split bx into these two factors multiplied by x Factor by grouping (always works)

5.3 Factoring Trinomials – Factor By Grouping Example: Split up and factor by grouping

5.3 More on Factoring Trinomials Factoring ax2 + bx + c by using FOIL (in reverse) The first terms must give a product of ax2 (pick two) The last terms must have a product of c (pick two) Check to see if the sum of the outer and inner products equals bx Repeat steps 1-3 until step 3 gives a sum = bx

5.3 More on Factoring Trinomials Example:

5.3 More on Factoring Trinomials Box Method (not in book):

5.3 More on Factoring Trinomials Box Method – keep guessing until cross-product terms add up to the middle value

5.4 Special Factoring Rules Difference of 2 squares: Example: Note: the sum of 2 squares (x2 + y2) cannot be factored.

5.4 Special Factoring Rules Perfect square trinomials: Examples:

5.4 Special Factoring Rules Difference of 2 cubes: Example:

5.4 Special Factoring Rules Sum of 2 cubes: Example:

5.4 Special Factoring Rules Summary of Factoring Factor out the greatest common factor Count the terms: 4 terms: try to factor by grouping 3 terms: check for perfect square trinomial. If not a perfect square, use general factoring methods 2 terms: check for difference of 2 squares, difference of 2 cubes, or sum of 2 cubes Can any factors be factored further?

5.5 Solving Quadratic Equations by Factoring Zero-Factor Property: If a and b are real numbers and if ab=0 then either a = 0 or b = 0

5.5 Solving Quadratic Equations by Factoring Solving a Quadratic Equation by factoring Write in standard form – all terms on one side of equal sign and zero on the other Factor (completely) Set all factors equal to zero and solve the resulting equations (if time available) check your answers in the original equation

5.5 Solving Quadratic Equations by Factoring Example:

5.6 Applications of Quadratic Equations This section covers applications in which quadratic formulas arise. Example: Pythagorean theorem for right triangles (see next slide)

5.6 Applications of Quadratic Equations Pythagorean Theorem: In a right triangle, with the hypotenuse of length c and legs of lengths a and b, it follows that c2 = a2 + b2 c a b

5.6 Applications of Quadratic Equations Example x+2 x x+1