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## 5.1 Factoring – the Greatest Common Factor

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**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: • 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**• Quadratic Equation: • Zero-Factor Property:If a and b are real numbers and if ab=0then 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