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5.1 Factoring – the Greatest Common Factor

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

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5.1 Factoring – the Greatest Common Factor

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  1. 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

  2. 5.1 Factoring – the Greatest Common Factor • Example: factor 5x2y + 25xy2z

  3. 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

  4. 5.1 Factoring – Factor By Grouping • Example:This arrangement doesn’t work. • Rearrange and try again

  5. 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

  6. 5.2 Factoring Trinomials • Example: • Example:

  7. 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)

  8. 5.3 Factoring Trinomials – Factor By Grouping • Example: • Split up and factor by grouping

  9. 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

  10. 5.3 More on Factoring Trinomials • Example:

  11. 5.3 More on Factoring Trinomials • Box Method (not in book):

  12. 5.3 More on Factoring Trinomials • Box Method – keep guessing until cross-product terms add up to the middle value

  13. 5.4 Special Factoring Rules • Difference of 2 squares: • Example: • Note: the sum of 2 squares (x2 + y2) cannot be factored.

  14. 5.4 Special Factoring Rules • Perfect square trinomials: • Examples:

  15. 5.4 Special Factoring Rules • Difference of 2 cubes: • Example:

  16. 5.4 Special Factoring Rules • Sum of 2 cubes: • Example:

  17. 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?

  18. 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

  19. 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

  20. 5.5 Solving Quadratic Equations by Factoring • Example:

  21. 5.6 Applications of Quadratic Equations • This section covers applications in which quadratic formulas arise.Example: Pythagorean theorem for right triangles (see next slide)

  22. 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

  23. 5.6 Applications of Quadratic Equations • Example x+2 x x+1

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