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Factoring

Center of Academic Assitance
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Polynomials
  • A term is the product of a number and/or a variable raised to a non-negative integer power, 3x4y, for example.  A polynomial is an expression consisting of one or more terms.
  •                                                          Examples:
  • A monomial has one term                           7
  • A binomial has two terms                     x2    +  3xy
  • A trinomial has three terms               3p   -  6q  +  r
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Degree of a Polynomial
  • The single largest total number of degrees (or powers) to which the variables in any one term are raised is called the degree of a polynomial.
  • Examples:
  • 2x3     degree = 3   variable x raised to the third power
  • 4x2y1   degree = 3   variable x raised to the second power plus   variable y raised to the first power
  • x4y3 - 4x2y1    degree = 7   variable x raised to the fourth power plus variable y raised to the third power                Note: the second term, 4x2y1, has degree = 3, since variable x is raised to the second power and variable y is raised to the first power
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Factoring a Polynomial
  • Remove common factors
  • Express the polynomial as a product of its factors


  •     Examples:
  •     Polynomial                    Factored Form
  •     2x3 + 8x2          2x2 (x + 4)
  •     3z5 + 3z4 - 12z - 12        3(z5 + z4 - 4z - 4)
  •       3(z + 1)(z2 + 2)(z2 - 2)
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Removal of Common Factors
  • For 6x2 + 12x - 3
  • Remove the common factor of 3:                     3(2x2 + 4x - 1)


  • For x2y3z6 - x4y3z4 + x2y5z4
  • Remove common factor of x2y3z4:
  •     x2y3z4(z2 - x2 + y2)
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Factoring Binomials
  • Difference of Squares
  • Difference of Cubes
  • Sum of Cubes
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Factoring Binomials: Difference of Squares
  •                     A2 - B2 = (A - B)(A + B)


  • Examples:
  • x2 - y2  = (x - y)(x + y)


  •     4r2 - 9s2  = (2r)2 - (3s)2
  •     = (2r - 3s)(2r + 3s)
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Factoring Binomials: Difference of Cubes
  •            A3 - B3  = (A - B)(A2 + AB + B2)


  • Examples:
  • x3 - y3 = (x - y)(x2 + xy + y2) cannot be further factored
  • 2z6 - 54 = 2(z6 - 27)
      •      = 2[(z2)3 - 33]
      •      = 2[(z2 - 3)(z4 + 3z2 + 32)]
      •       = 2(z2 - 3)(z4 + 3z2 + 9)
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Factoring Binomials:
Sum of Cubes
  •     A3 + B3 = (A + B)(A2 - AB + B2)


  • Examples:
  • x3  + y3  = (x + y)(x2 - xy + y2)
  • x9 + 1 = (x3)3 + 13
  • = (x3 + 1)[(x3)2 - x3 + 1)]
  • = (x3 + 1)(x6 - x3 + 1)
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Factoring Trinomials
  • For the common form of the trinomial     ax2  + bx + c
  • let a1 and a2 be factors of  a
  • let s1 and s2 be factors of  c
  • let  b  be expressed as  a1s2 + a2s1


  • Then the factored form of the trinomial becomes
  •                             (a1x + s1)(a2x + s2)
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Factoring Monic Quadratic Trinomials
  • In the monic form of the trinomial  ax2  + bx + c, the
  • value of  a  is one.  The factored form is represented by
  • (x + s1)(x + s2).
  • a1a2 = 1,  s1 + s2 = b,  s1s2 = c
  • Example:
  • For the trinomial  x2 + 3x - 40        a = 1, b = 3, c = -40
  • s1 + s2 = 3
  • s1s2 = -40
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Finding s1 and s2
  • Example:
  • For the trinomial  x2 + 3x - 40        a = 1, b = 3, c = -40
  • The factors of -40 are
  • -1 and 40 1 and -40
  • -2 and 20 2 and -20
  • -4 and 10 4 and -10
  • -8 and 5 8 and -5
  • Test each pair of factors.  One set will sum to 3.  Use
  • that set to substitute for s1 and s2 in the factored form
  • for the trinomial.
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Finding s1 and s2  (cont)
  • Example:
  • For the trinomial  x2 + 3x - 40      a = 1, b = 3, c = -40
  • The pair of factors which sum to 3 are 8 and -5.
  • The factored form for the trinomial becomes            (x + 8)(x - 5)
  • Multiply out to check:
  •     x2 - 5x + 8x - 40
  •         x2 + 3x - 40
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Monic Quadratics
  • Not every monic quadratic trinomial can be factored


  • You may need to use the quadratic formula
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Factoring Nonmonic Quadratic Trinomials ax2 + bx + c
  • If possible, express ac as a product of two integers
  • that have a sum of b.
  • Example:
  • For the trinomial  6x2 - 19x + 15    a = 6, b = -19, c = 15
  •                                                                  and  ac = 90
  • By trial and error, find two integers
  • whose product = 90 and sum = -19 …
  •     ... the two integers are -9 and -10
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Factoring  ax2 + bx + c  (cont)
  • Example:
  • For the trinomial  6x2 - 19x + 15      a = 6, b = -19, c = 15
  •                                                                           and  ac = 90
  • Rewrite the term bx using the two integers  -9 and -10               6x2 + (-9x -10x) + 15
  • Group the 4 terms into 2 pairs
  •     (6x2 - 9x) + (-10x + 15)
  • Remove the common factors from each pair
  •     3x(2x - 3) - 5(2x - 3)
  • Distribute the term that is a common factor
  •     (2x - 3)(3x - 5)
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Factoring Polynomials With Four Terms by Grouping
  • Example:
  • Factor the polynomial   x3 + x2 - x - 1
  • Group into pairs and remove common factors
    •    (x3 + x2) + (-x - 1)
    •     x2(x + 1) + (-1)(x + 1)
    •     x2(x + 1) - 1(x + 1)
  • Distribute the term that is a common factor
    •      x2(x + 1) - 1(x + 1)
    •      (x + 1)(x2 - 1)
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Algebra Review Homework 1
  • Example:
  • x4y + 27xy4


  • common factor xy
  • xy (x3 + 27y3)
  • (x3 + 27y3) is a sum of cubes
  • sum of cubes where  A =  x and  B = 3y
  • xy [x3 + (3y)3]
  • xy (x + 3y) (x2 - 3xy + 9y2 )
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Algebra Review Homework 2
  • Example:
  • q6 - 1


  • difference of squares
  • ( q3)2 - 12
  • (q3 - 1) (q3 + 1)


  •  (q3 - 1) difference of cubes   (q3 + 1) sum of cubes
  •  (q - 1) (q2 + q + 1)       (q + 1) (q2 - q + 1)