Factoring
Factoring
Multiplying Binomials (FOIL)              Multiply. (x+3)(x+2)Distribute.    x•x+x•2+3•x+3•2                F   O   I   L ...
Multiplying Binomials (Tiles)                   Multiply. (x+3)(x+2)Using Algebra Tiles, we have:               x +       ...
Factoring Trinomials (Tiles)                How can we factor trinomials such as                 x2 + 7x + 12 back into bi...
Factoring Trinomials (Tiles)                How can we factor trinomials such as                 x2 + 7x + 12 back into bi...
Factoring Trinomials (Tiles)                How can we factor trinomials such as                 x2 + 7x + 12 back into bi...
Factoring Trinomials (Tiles)                How can we factor trinomials such as                 x2 + 7x + 12 back into bi...
Factoring Trinomials (Tiles)              How can we factor trinomials such as               x2 + 7x + 12 back into binomi...
Factoring Trinomials (Method 2)         Again, we will factor trinomials such as            x2 + 7x + 12 back into binomia...
Factoring Trinomials (Method 2)                     x2 + 7x + 12  Step 2: Choose the pair that      12 = 1 • 12  adds up t...
Factoring Trinomials (Method 2)                    Factor.    x2 + 2x - 24               This time, the constant is negati...
Factoring Trinomials (Method 2*)                      Factor. 3x2 + 14x + 8   This time, the x2 term DOES have a coefficie...
Factoring Trinomials (Method 2*)                   Factor. 3x2 + 14x + 8 Step 4: Write temporary            ( x + 2 )( x +...
Factoring Trinomials (Method 2*)                  Factor. 3x2 + 14x + 8 You should always check the factors by distributin...
Factoring Trinomials (Method 2*)                       Factor 3x2 + 11x + 4   This time, the x2 term DOES have a coefficie...
Factor These Trinomials!Factor each trinomial, if possible. The first four do NOT haveleading coefficients, the last two D...
Solution #1:                   t2 – 4t – 211) Factors of -21:       1 • -21, -1 • 21                         3 • -7, -3 • ...
Solution #2:                   x2 + 12x + 321) Factors of 32:         1 • 32                          2 • 16              ...
Solution #3:                        x2 - 10x + 241) Factors of 32:          1 • 24           -1 • -24                     ...
Solution #4:                    x2 + 3x - 181) Factors of -18:      1 • -18, -1 • 18                        2 • -9, -2 • 9...
Solution #5:                      2x2 + x - 211) Multiply 2 • (-21) = - 42;     1 • -42, -1 • 42   list factors of - 42.  ...
Solution #6:                    3x2 + 11x + 101) Multiply 3 • 10 = 30;          1 • 30   list factors of 30.            2 ...
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Factoring notes

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Factoring notes

  1. 1. Factoring
  2. 2. Factoring
  3. 3. Multiplying Binomials (FOIL) Multiply. (x+3)(x+2)Distribute. x•x+x•2+3•x+3•2 F O I L = x2+ 2x + 3x + 6 = x2+ 5x + 6
  4. 4. Multiplying Binomials (Tiles) Multiply. (x+3)(x+2)Using Algebra Tiles, we have: x + 3 x x2 x x x + = x2 + 5x + 6 x 1 1 1 2 x 1 1 1
  5. 5. Factoring Trinomials (Tiles) How can we factor trinomials such as x2 + 7x + 12 back into binomials?One method is to again use algebra tiles:1) Start with x2. x2 x x x x x2) Add seven “x” tiles(vertical or horizontal, at x 1 1 1 1 1least one of each) andtwelve “1” tiles. x 1 1 1 1 1 1 1
  6. 6. Factoring Trinomials (Tiles) How can we factor trinomials such as x2 + 7x + 12 back into binomials?One method is to again use algebra tiles:1) Start with x2. x2 x x x x x2) Add seven “x” tiles(vertical or horizontal, at x 1 1 1 1 1least one of each) andtwelve “1” tiles. x 1 1 1 1 13) Rearrange the tiles 1 1until they form a We need to change the “x” tiles sorectangle! the “1” tiles will fill in a rectangle.
  7. 7. Factoring Trinomials (Tiles) How can we factor trinomials such as x2 + 7x + 12 back into binomials?One method is to again use algebra tiles:1) Start with x2. x2 x x x x x x2) Add seven “x” tiles(vertical or horizontal, at x 1 1 1 1 1 1least one of each) andtwelve “1” tiles. 1 1 1 1 1 13) Rearrange the tilesuntil they form a Still not a rectangle.rectangle!
  8. 8. Factoring Trinomials (Tiles) How can we factor trinomials such as x2 + 7x + 12 back into binomials?One method is to again use algebra tiles:1) Start with x2. x2 x x x x2) Add seven “x” tiles(vertical or horizontal, at x 1 1 1 1least one of each) andtwelve “1” tiles. x 1 1 1 1 x 1 1 1 13) Rearrange the tilesuntil they form arectangle! A rectangle!!!
  9. 9. Factoring Trinomials (Tiles) How can we factor trinomials such as x2 + 7x + 12 back into binomials?One method is to again use algebra tiles:4) Top factor: x + 4The # of x2 tiles = x’s x x2 x x x xThe # of “x” and “1”columns = constant. + x 1 1 1 1 3 x 1 1 1 15) Side factor:The # of x2 tiles = x’s x 1 1 1 1The # of “x” and “1”rows = constant. x2 + 7x + 12 = ( x + 4)( x + 3)
  10. 10. Factoring Trinomials (Method 2) Again, we will factor trinomials such as x2 + 7x + 12 back into binomials. This method does not use tiles, instead we look for the pattern of products and sums! If the x2 term has no coefficient (other than 1)... x2 + 7x + 12 Step 1: List all pairs of 12 = 1 • 12 numbers that multiply to =2•6 equal the constant, 12. =3•4
  11. 11. Factoring Trinomials (Method 2) x2 + 7x + 12 Step 2: Choose the pair that 12 = 1 • 12 adds up to the middle =2•6 coefficient. =3•4 Step 3: Fill those numbers into the blanks in the ( x + 3 )( x + 4 ) binomials: x2 + 7x + 12 = ( x + 3)( x + 4)
  12. 12. Factoring Trinomials (Method 2) Factor. x2 + 2x - 24 This time, the constant is negative!Step 1: List all pairs of -24 = 1 • -24, -1 • 24numbers that multiply to equalthe constant, -24. (To get -24, = 2 • -12, -2 • 12one number must be positive and = 3 • -8, -3 • 8one negative.) = 4 • -6, - 4 • 6Step 2: Which pair adds up to 2?Step 3: Write the binomial x2 + 2x - 24 = ( x - 4)( x + 6)factors.
  13. 13. Factoring Trinomials (Method 2*) Factor. 3x2 + 14x + 8 This time, the x2 term DOES have a coefficient (other than 1)! Step 1: Multiply 3 • 8 = 24 24 = 1 • 24 (the leading coefficient & constant). = 2 • 12 Step 2: List all pairs of =3•8 numbers that multiply to equal that product, 24. =4•6 Step 3: Which pair adds up to 14?
  14. 14. Factoring Trinomials (Method 2*) Factor. 3x2 + 14x + 8 Step 4: Write temporary ( x + 2 )( x + 12 ) factors with the two numbers. 3 3 Step 5: Put the original 4 leading coefficient (3) under ( x + 2 )( x + 12 ) both numbers. 3 3 Step 6: Reduce the fractions, if ( x + 2 )( x + 4 ) possible. 3 Step 7: Move denominators in ( 3x + 2 )( x + 4 ) front of x.
  15. 15. Factoring Trinomials (Method 2*) Factor. 3x2 + 14x + 8 You should always check the factors by distributing, especially since this process has more than a couple of steps. ( 3x + 2 )( x + 4 ) = 3x • x + 3x • 4 + 2 • x + 2 • 4 = 3x2 + 14 x + 8 √ 3x2 + 14x + 8 = (3x + 2)(x + 4)
  16. 16. Factoring Trinomials (Method 2*) Factor 3x2 + 11x + 4 This time, the x2 term DOES have a coefficient (other than 1)! Step 1: Multiply 3 • 4 = 12 12 = 1 • 12 (the leading coefficient & constant). =2•6 Step 2: List all pairs of numbers that multiply to equal =3•4 that product, 12. Step 3: Which pair adds up to 11? None of the pairs add up to 11, this trinomial can’t be factored; it is PRIME.
  17. 17. Factor These Trinomials!Factor each trinomial, if possible. The first four do NOT haveleading coefficients, the last two DO have leading coefficients.Watch out for signs!! 1) t2 – 4t – 21 2) x2 + 12x + 32 3) x2 –10x + 24 4) x2 + 3x – 18 5) 2x2 + x – 21 6) 3x2 + 11x + 10
  18. 18. Solution #1: t2 – 4t – 211) Factors of -21: 1 • -21, -1 • 21 3 • -7, -3 • 72) Which pair adds to (- 4)?3) Write the factors. t2 – 4t – 21 = (t + 3)(t - 7)
  19. 19. Solution #2: x2 + 12x + 321) Factors of 32: 1 • 32 2 • 16 4•82) Which pair adds to 12 ?3) Write the factors. x2 + 12x + 32 = (x + 4)(x + 8)
  20. 20. Solution #3: x2 - 10x + 241) Factors of 32: 1 • 24 -1 • -24 2 • 12 -2 • -12 3•8 -3 • -8 4•6 -4 • -62) Which pair adds to -10 ? None of them adds to (-10). For the numbers to multiply to +24 and add to -10, they must both be negative!3) Write the factors. x2 - 10x + 24 = (x - 4)(x - 6)
  21. 21. Solution #4: x2 + 3x - 181) Factors of -18: 1 • -18, -1 • 18 2 • -9, -2 • 9 3 • -6, -3 • 62) Which pair adds to 3 ?3) Write the factors. x2 + 3x - 18 = (x - 3)(x + 18)
  22. 22. Solution #5: 2x2 + x - 211) Multiply 2 • (-21) = - 42; 1 • -42, -1 • 42 list factors of - 42. 2 • -21, -2 • 21 3 • -14, -3 • 142) Which pair adds to 1 ? 6 • -7, -6 • 73) Write the temporary factors. ( x - 6)( x + 7) 2 24) Put “2” underneath. 3 ( x - 6)( x + 7)5) Reduce (if possible). 2 26) Move denominator(s)in ( x - 3)( 2x + 7)front of “x”. 2x2 + x - 21 = (x - 3)(2x + 7)
  23. 23. Solution #6: 3x2 + 11x + 101) Multiply 3 • 10 = 30; 1 • 30 list factors of 30. 2 • 15 3 • 102) Which pair adds to 11 ? 5•63) Write the temporary factors. ( x + 5)( x + 6) 3 34) Put “3” underneath. 2 ( x + 5)( x + 6)5) Reduce (if possible). 3 36) Move denominator(s)in ( 3x + 5)( x + 2)front of “x”. 3x2 + 11x + 10 = (3x + 5)(x + 2)

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