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Section 5.4 Factoring

Section 5.4 Factoring. FACTORING Greatest Common Factor, Factor by Grouping, Factoring Trinomials, Difference of Squares, Perfect Square Trinomial, Sum & Difference of Cubes. Factoring—define factored form. Factor means to write a quantity as a multiplication problem

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Section 5.4 Factoring

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  1. Section 5.4 Factoring FACTORING Greatest Common Factor, Factor by Grouping, Factoring Trinomials, Difference of Squares, Perfect Square Trinomial, Sum & Difference of Cubes

  2. Factoring—define factored form • Factor means to write a quantity as a multiplication problem • a product of the factors. • Factored forms of 18 are:

  3. Factoring: The Greatest Common Factor • To find the greatest common factor of a list of numbers: • Write each number in prime factored form • Choose the least amount of each prime that occurs in each number • Multiply them together • Find the GCF of 24 & 36

  4. Factoring: The Greatest Common Factor • To find the greatest common factor of a list of variable terms: • Choose the variables common to each term. • Choose the smallest exponent of each common variable. • Multiply the variables. • Find the GCF of:

  5. Factoring: The Greatest Common Factor • To factor out the greatest common factor of a polynomial: • Choose the greatest common factor for the coefficients. • Choose the greatest common factor for the variable parts. • Multiply the factors.

  6. Factoring: The Greatest Common Factor • Factor each of the following by factoring out the greatest common factor: 5x + 5 = 4ab + 10a2 = 8p4q3 + 6p3q2 = 2y + 4y2 + 16y3 = 3x(y + 2) -1(y + 2) =

  7. Factoring: The Greatest Common Factor • The answers are :

  8. Factoring: by Grouping • Often used when factoring four terms. • Organize the terms in two groups of two terms. • Factor out the greatest common factor from each group of two terms. • Factor out the common binomial factor from the two groups. • Rearranging the terms may be necessary.

  9. Factoring: by Grouping • Factor by grouping: • 2 groups of 2 terms • Factor out the GCF from each group of 2 terms • Factor out the common binomial factor

  10. Factoring: by Grouping Factor by grouping

  11. Factoring Trinomials—with a coefficient of 1 for the squared term • Factor: • List the factors of 20: • Select the pairs from which 12 may be obtained • Write the two binomial factors: • Check using FOIL:

  12. Factoring Trinomials TIP  If the last term of the trinomial is positive and the middle sign is positive, both binomials will have the same “middle” sign as the second term.

  13. Factoring Trinomials TIP  If the last term of the trinomial is positive and the middle sign is negative, both binomials will have the same “middle” sign as the second term.

  14. Factoring Trinomials—with a coefficient of 1 for the squared term • Factor • List the factors of 22 • Select the pair from which –9 may be obtained • Write the two binomial factors: • Check using FOIL:

  15. Factoring Trinomials TIP  If the last term of the trinomial is negative, both binomials will have one plus and one minus “middle” sign.

  16. Factoring Trinomials—primes • A PRIME POLYNOMIAL cannot be factored using only integer factors. • Factor : • The factors of 5: 1 and 5. • Since –2 cannot be obtained from 1 and 5, the polynomial is prime.

  17. Factoring Trinomials—2 variables • Factor: • The factors of 8 are: 1,8 & 2,4, & -1,-8 & -2, -4 • Choose the pairs from which –6 can be obtained: 2 & 4 • Use y in the first position and z in the second position • Write the two binomial factors and check your answer

  18. Factoring Trinomials—with a GCF • If there is a greatest common factor? • If yes, factor it out first.

  19. Always check your answer with multiplication of the factors. The check: Factoring Trinomials—always check your factored form

  20. Factoring Trinomials—when the coefficient is not 1 on the squared term

  21. Factoring Trinomials---use grouping

  22. Factoring Trinomials---use grouping

  23. Factoring Trinomials---use FOIL and Trial and Error

  24. Factoring Trinomials---use FOIL and Trial and Error

  25. Factoring Trinomials---use FOIL and Trial and Error

  26. Factoring Trinomials---use FOIL and Trial and Error

  27. Factoring Trinomials---use FOIL and Trial and Error

  28. Factoring Trinomials---use FOIL and Trial and Error

  29. Factoring Trinomials---with a negative GCF • Is the squared term negative? • If yes, factor our a negative GCF.

  30. Special Factoring—difference of 2 squares • The following must be true: • There must be only two terms in the polynomial. • Both terms must be perfect squares. • There must be a “minus” sign between the two terms.

  31. Special Factoring—difference of 2 squares • The following pattern holds true for the difference of 2 squares:

  32. Special Factoring—difference of 2 squares • The pattern:

  33. Special Factoring—difference of 2 squares • The pattern:

  34. Special Factoring—difference of 2 squares • The pattern:

  35. Special Factoring—difference of 2 squares • The pattern:

  36. Special Factoring—perfect square trinomial • A perfect square trinomial is a trinomial that is the square of a binomial.

  37. Special Factoring—perfect square trinomial • The first and third terms are perfect squares. • AND the middle term is twice the product of the square roots of the first and third terms • TEST THE MIDDLE TERM:

  38. Special Factoring—perfect square trinomial • The patterns for a perfect square trinomial are:

  39. Special Factoring—perfect square trinomial • Factor the following using the perfect square trinomial pattern:

  40. Special Factoring—perfect square trinomial • Factor the following using the perfect square trinomial pattern:

  41. Special Factoring—difference of two cubes • Factor using the pattern.

  42. Special Factoring—sum of two cubes • Factor using the pattern.

  43. Solving quadratic equation with factoring • A quadratic equation has a “squared” term.

  44. ZERO FACTOR PROPERTY To Factor a Quadratic, Apply the Zero-Factor Property. • If a and b are real numbers and if ab = 0, then a = 0 or b = 0.

  45. Solving quadratic equations with factoring—Zero-Factor Property • Solve the equation: (x + 2)(x - 8) = 0. • Apply the zero-factor property: (x + 2) = 0 or (x – 8) = 0 x = -2 or x = 8

  46. Solving quadratic equations with factoring—Zero-Factor Property • There are two answers for x: -2 and 8. • Check by substituting the values calculated for x into the original equation. (x + 2)(x - 8) = 0. (-2 + 2)(-2 – 8) = 0 (8 + 2)(8 – 8) = 0 0 = 0 0 = 0

  47. Solving quadratic equations with factoring—Standard Form • To solve a quadratic equation, • Write the equation in standard form. • (Solve the equation for 0.)

  48. Solving quadratic equations with factoring • To solve a quadratic equation, • Factor the quadratic expression.

  49. Solving quadratic equations with factoring • To solve a quadratic equation, • Apply the Zero-Factor Property

  50. Solving quadratic equations with factoring • To solve a quadratic equation, • Check your answers

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