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Factoring Polynomials

Factoring Polynomials. Geogebra Efil Mileny Catayong. Common Monomial Factor. Factoring the GCF from Polynomials. Review Algebraic Factorization is the writing of an expression as the product of prime numbers and variables with no variables having an exponent greater than one.

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Factoring Polynomials

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  1. Factoring Polynomials Geogebra Efil MilenyCatayong

  2. Common Monomial Factor

  3. Factoring the GCF from Polynomials • Review AlgebraicFactorization is the writing of an expression as the product of prime numbers and variables with no variables having an exponent greater than one. Example: 14x2y = 2  7  x  x  y

  4. Factoring the GCF from Polynomials • Write each term on a separate line and then write the algebraic factorization of each term. • “Pair up” all factors that occur in each term and circle them.

  5. 3x3 = 3  x  x  x • 6x2y = 2  3  x  x  y • 15xy2 = 3  5  x  y  y • Step 1 Example: FACTOR:3x3 + 6x2y– 15xy2 3.Multiply what is circled. This is the GCF. When writing the answer, put it outside parenthesis. The “left-overs” for each term should be multiplied and put inside parenthesis.

  6. Step 3 • Step 4 • 3x3 = 3  x  x  x • 6x2y = 2  3  x  x  y • 15xy2 = 3  5  x  y  y • 3x (x2 + 2xy – 5y2)

  7. Try these yourself. • 15x2 – 20xy • 3x2 + 15x • 20abc + 15a2c – 5ac

  8. Factoring Trinomials

  9. Distribute. x • x + x • 2 + 3 • x + 3 • 2 F O I L = x2+ 2x + 3x + 6 = x2+ 5x + 6 Review Multiplying Binomials (FOIL) Multiply. (x+3)(x+2)

  10. x2 x x x x x 1 1 1 1 1 1 Review Multiplying Binomials (Tiles) Multiply. (x+3)(x+2) Using Algebra Tiles, we have: x + 3 x + 2 = x2 + 5x + 6

  11. x2 x x x x x x x 1 1 1 1 1 1 1 1 1 1 1 1 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. 2) Add seven “x” tiles (vertical or horizontal, at least one of each) and twelve “1” tiles.

  12. x2 x x x x x x x 1 1 1 1 1 1 1 1 1 1 1 1 Factoring Trinomials (Tiles) 1) Start with x2. 2) Add seven “x” tiles (vertical or horizontal, at least one of each) and twelve “1” tiles. 3) Rearrange the tiles until they form a rectangle! Still not a rectangle.

  13. x2 x x x x x x x 1 1 1 1 1 1 1 1 1 1 1 1 Factoring Trinomials (Tiles) 1) Start with x2. 2) Add seven “x” tiles (vertical or horizontal, at least one of each) and twelve “1” tiles. 3) Rearrange the tiles until they form a rectangle! A rectangle!!!

  14. x2 x x x x x x x 1 1 1 1 1 1 1 1 1 1 1 1 Factoring Trinomials (Tiles) 4) Top factor:The # of x2 tiles = x’sThe # of “x” and “1” columns = constant. + 4 x 5) Side factor:The # of x2 tiles = x’sThe # of “x” and “1” rows = constant. x + 3 x2 + 7x + 12 = ( x + 4)( x + 3)

  15. Factoring Trinomials (Method 2) Again, we will factor trinomials such as x2 + 7x + 12 back into binomials. If the x2 term has no coefficient (other than 1)... x2 + 7x + 12 Step 1: List all pairs of numbers that multiply to equal the constant, 12. 12 = 1 • 12 = 2 • 6 = 3 • 4

  16. 12 = 1 • 12 = 2 • 6 = 3 • 4 ( x + )( x + ) 4 3 x2 + 7x + 12 Step 2: Choose the pair that adds up to the middle coefficient. Step 3: Fill those numbers into the blanks in the binomials: x2 + 7x + 12 = ( x + 3)( x + 4)

  17. Factor These Trinomials! Factor each trinomial, if possible. The first four do NOT have leading 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. Perfect Square Trinomials • When factoring using perfect square trinomials, look for the following three things: • 3 terms • last term must be positive • first and last terms must be perfect squares • If all three of the above are true, write one ( )2 using the sign of the middle term.

  19. Try These • 1. a2 – 8a + 16 • 2. x2 + 10x + 25 • 3. 4y2 + 16y + 16 • 4. 9y2 + 30y + 25 • 5. 3r2 – 18r + 27 • 6. 2a2 + 8a - 8

  20. Difference of Squares

  21. Difference of Squares • When factoring using a difference of squares, look for the following three things: • only 2 terms • minus sign between them • both terms must be perfect squares

  22. Difference of Squares • If all 3 of the above are true, write two ( ), one with a + sign and one with a – sign : ( + ) ( - ).

  23. Try These • 1. a2 – 8a + 16 • 2. x2 + 10x + 25 • 3. 4y2 + 16y + 16 • 4. 9y2 + 30y + 25 • 5. 3r2 – 18r + 27 • 6. 2a2 + 8a - 8

  24. Factoring Four Term Polynomials

  25. Factor by Grouping • When polynomials contain four terms, it is sometimes easier to group like terms in order to factor. • Your goal is to create a common factor. • You can also move terms around in the polynomial to create a common factor. • Practice makes you better in recognizing common factors.

  26. Factor by GroupingExample • FACTOR:3xy - 21y + 5x – 35 • Factor the first two terms: 3xy - 21y= 3y(x – 7) • Factor the last two terms: + 5x - 35 = 5(x – 7) • Thewhiteparentheses are the same so it’s the common factor Now you have a common factor (x - 7)(3y + 5)

  27. Factoring Completely

  28. Factoring Completely • Now that we’ve learned all the types of factoring, we need to remember to use them all. • Whenever it says to factor, you must break down the expression into the smallest possible factors. • Let’s review all the ways to factor.

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