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5.4 – Factoring Polynomials Completely

Key Terms

  • Factor Theorem – A number a is a root of a polynomial function if and only if, when dividing the polynomial by (x – a), the remainder equals zero.
    • If you get a remainder when you divide polynomials, you do NOT have a root.
  • Rational Root Theorem – The rational roots of a polynomial function F(x) can be written in the form \frac{p}{q}, where p is a factor of the constant term of the polynomial, and q is a factor of the leading coefficient.
  • Remainder Theorem – A theorem stating that when the polynomial in F(x) is divided by x – a, the remainder is F(a).
    • Also, when F(x) is divided by x + a, the remainder is F(-a).

 


Review

  • Divisibility Rules

Alg2B 5.4 Divisibility Rules

 

  • Roots
    • Roots are the “zeros” of factors.
    • Roots  are the x-intercepts of the graph.
    • To find the root of a factor, set the factor equal to zero.
      • Ex. F(x)=x^2+3x+2 has factors of (x+1) and (x+2)
      • Set the factors to equal zero and solve for x:  x+1=0 and x+2=0
      • For the first factor: (x + 1), subtract 1 on both sides:  x = -1
      • For the second factor (x + 2), subtract 2 on both sides: x = -2
      • Lastly, test the roots using substitution.
      • For F(-1)
        • F(x)=x^2+3x+2
        • F(-1)=(-1)2+3(-1)+2
        • F(-1)=1-3+2
        • F(-1)=0
        • So, -1 IS a root of the polynomial.
      • For F(-2)
        • F(x)=x^2+3x+2
        • F(-2)=(-2)2+3(-2)+2
        • F(-2)=-4-6+2
        • F(-2)=0
        • So, -2 IS a root of the polynomial.

 

  • Roots and Factors
    • Roots have the opposite signs from their factors
      • If (x + 4) is a factor, then x = -4 is a root.
      • If (x – 6) is a factor, then x = 6 is a root.

 

  • Factoring
    • The constant term of a polynomial may have positive or negative factors
    • The x-term of a polynomial is the sum of the factors of the constant term (which may be positive or negative).

Alg2B 5.4 Factoring Ex

 


Notes

  • Factors, Roots, and Remainders
    • A number a is a root of P(x) if and only if the remainder, when dividing the polynomial by x – a, equals zero.
    • If a polynomial is divided by (x – a) and the remainder equals zero, then (x – a) is a factor of the polynomial.

 

  • Roots
    • If the leading coefficient is 1 (problem 2), just find all the factors of the constant term.
    • If the leading coefficient is not 1 (problem 1), use the rational root theorem.
      • The rational roots can be written as \frac{p}{q}
      • p is a factor of the constant term
      • q is a factor of the leading coefficient

 

  • Coefficient of 1
    • Example: F(x)=x^2-5+14
    • Possible roots are: \pm 1, \pm 2, \pm 7, \pm 14

 

  • Coefficient of a Number Other than 1
    • Example:  F(x)=3x^2-5x-2

Alg2B 5.4 Rational Root Ex

 


Alg2B 5.4 Radical Roots

 


  • Conjugates
    • How do the three rules below help you check your answers to polynomial equations?
      • You know the maximum number of answers you should have.
      • You know that for every radical and complex answer, you should also have its conjugate as another answer.

Alg2B 5.4 Conjugates

 

  • Complex Roots
    • For graphs, there are NO x-intercepts for complex roots.

 


  • Factoring Polynomials
    • You only need to find one root!
    • You don’t have to test all the possible roots with synthetic division.
    • When you find one that works (in other words, when the remainder is zero), move on to step 3
    • To begin factoring, start with the roots 1, then try -1

Alg2B 5.4 Factoring Poly

 


Examples

  • Example 1
    • What is a root (or zero) of a polynomial function?
    • Answer: A value of the variable that makes the polynomial equal to zero

 

  • Example 2
    • -6 is NOT a root because there is a remainder of 11
    • (x + 6) is NOT a factor because there is a remainder of 11
    • When x = -6, 2x^2+9x-7=11
    • When 2x^2+9x-7 is divided by x+6, the remainder is 11Alg2B 5.4 Roots Factors

 

  • Example 3
    • 4 is a root of F(x)=3x^2-13x+4
    • (x-4) is a factor of 3x^2-13x+4
    • (3x^2-13x+4) ÷ (x-4) = (3x-1)
    • (3x^2-13x+4) ÷ (3x-1) = (x-4)
    • The remainder is zero

Alg2B 5.4 Roots Factors Ex2

 


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