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3.2 – ax^2+bx+c

Objectives

  • List the steps used to factor a trinomial that has a leading coefficient other than 1.
  • Find the factors of a trinomial that has a leading coefficient that is not 1.
  • Find the trinomial for a given set of factors.

 

Key Terms

  • Common Factor – A number or expression that is a factor of two or more numbers or polynomials.
  • Factor It Out – To write an expression as the product of a factor and another expression. The factor must be common to each term of the original expression. To factor out a common factor from an expression is equivalent to using the distributive property in reverse.
  • Factorization – The result of writing a number or expression as a product of two or more factors.
  • Reducible Trinomial – A polynomial that has exactly three terms and is able to be factored.
    • Any reducible trinomial can be factored as  (rx\pm p)(sx\pm q), where r•s is the trinomial’s leading coefficient and p•q is the constant term

Alg 2A 03.02 - Factoring Coef04

Notes

Greatest Common Factor (GCF)
  • Rule: When the leading coefficient is a number other than 1, the first step is to look for a common factor in each term and factor it out.

Alg 2A 03.02 - Factoring Coef01

Factoring ANY Quadratic Trinomial
Alg2A 03.02 - Factoring Rules 07


  • Trinomials
    • If you multiply the x-coefficients of the factors, you get the leading coefficient of the trinomial
    • If you multiply the constant terms of the factors, you get the constant term of the trinomial
    • The middle term depends on the factors and the signs.  You will need to use “trial and error” with the OI in FOIL.
    • FOIL: First, Outer, Inner, Last
      • FOIL is a form of distribution

 

  • To factor  ax^2 + bx + c, you must find all possible values of r, s, p, and q
    • Make two lists:
      • List 1: r and s
        • These are the factors of the leading coefficient
      • List 2: p and q
        • These are the factors of the constant term
Examples
  •  Ex 1:  8x^2+12x+4

Alg 2A 03.02 - Factor PQ

    • Use Trial and Error methods to find the right solution(s)
      • Try all the possibilities and see which ones work
      • Both of the following factorizations work:
        • First possibility: (4x + 4)(2x + 1)
          • Factor out the 4 from the first binomial: 4(x + 4)
          • 4(x + 1)(2x + 1)
        • Second possibility: (2x + 2)(4x + 2)
          • Factor out a 2 from each binomial: 2(x + 2) and 2(2x + 1)
          • Since 2*2 = 4, put the 4 in the front: 4(x + 1)(2x + 1)

 


  • Ex 2

Alg 2A 03.02 - Factoring Coef03

Alg 2A 03.02 - Factoring Coef02

 

Alg 2A 03.02 - Factoring Coef


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