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5.5 – Solving Polynomial Equations

Key Terms

  • Complex Conjugate Theorem – If a + bi is a root, then a – bi is also a root.
  • Conjugate Radical Theorem – If a+ \sqrt{b} is a root, then a- \sqrt{b} is also a root.
  • Fundamental Theorem of Algebra – A polynomial of degree n has exactly n complex roots.
  • Lower Bound – A value that is less than the smallest root of a polynomial.
  • Multiplicity – A count of the number of times a specific value occurs as a root of a polynomial function.
  • Upper Bound – A value that is larger than the largest root of a polynomial.

 


Review

  • Leading Coefficients
    • If the leading coefficient has no sign, it is a positive number.

 

  • Complex Numbers
    • Square roots of negative numbers produce complex numbers, which are written in the form a + bi (the combination of a real and an imaginary number).

 


Notes

  • Roots
    • If a solution set has imaginary or irrational roots, those numbers will not be listed when you use the rational root theorem: \frac{p}{q}.

 

  • Fundamental Theorem of Algebra
    • The degree of a polynomial function tells you how many roots, or zeros, it has!

 

  • Multiplicity
    • When you factor a polynomial like x^2-10x+25, you get two identical factors: (x − 5) and (x − 5); so, there are two identical roots: 5 and 5.
      • This root touches the x-axis once, at (5, 0); and, it is called a double root since there are two of the same.
    • Multiplicities must be included when you say that a polynomial of degree n has n roots.

 

  • Complex Roots and Imaginary Numbers
    • Complex Conjugate Theorem
      • Complex roots and imaginary roots always come in pairs.
      • If a + bi is a root, then a − bi is also a root.
    • When graphed, imaginary roots do not cross the x-axis.
    • Imaginary (complex) numbers cannot be graphed.

Alg2B 5.5 Complex Roots

Alg2B 5.5 Complex Roots2

 

  • Conjugate Radicals – the two roots:  a+ \sqrt{b} and a- \sqrt{b}

 

  • Possible Roots
    • If a polynomial has a degree of three, it can have 1 real root AND 2 complex roots or it may have 3 real roots.

 


  • Descartes’ Rule of Signs
    • The number of positive roots (or zeros) is less than or equal to the number of sign changes for the terms of a polynomial function, f(x).
    • The number of negative roots (or zeros) is less than or equal to the number of sign changes for the terms of a polynomial function, f(−x).
    • In each case, if the number of roots is less than the number of sign changes, then the number of roots differs from the number of sign changes by a multiple of 2.

Alg2B 5.5 Descartes Rule2

 

  • Example: Count the number of times the term changes from positive to negative or vice-versa
    • f(x)=x^3+x^2-x+2 changes from positive to negative, then back to positive
      • Two sign changes: + to – and then – to +
      • There will either be 2 or 0 positive roots (we know this because we tested a positive x in the function: f(x))
    • f(-x)=(-x)^3+(-x)^2-(-x)+2 simplifies to: f(x)=-x^3+x^2+x+2, which changes from negative to positive
      • Just one sign change: – to +
      • There will be 1 negative root (we know this because we tested a negative x in the function: f(-x))

Alg2B 5.5 Descartes Rule-x

 


  • Bounds
    • Upper bound: A number greater than or equal to all real roots.
      • Upper Bound Theorem: for a positive number c, if f(x) is divided by (x − c) and the resulting quotient polynomial and remainder have no changes in sign (all positive terms or all negative terms), then f(x) has no real roots greater than c.
    • Lower bound: A number less than or equal to all real roots.
      • Lower Bound Theorem: for a negative number c, if f(x) is divided by (x − c) and the resulting quotient and remainder have alternating signs, then f(x) has no real roots less than c.
    • Each type of bound has a specific pattern in synthetic division.
      • If synthetic division by a possible root gives NO changes in sign on the bottom row, then this number is the upper bound; and, any numbers larger than this number can be deleted from the list of possible roots.

 


  • Steps for Solving a Polynomial Equation
    1. Use the fundamental theorem of algebra to identify the total number of roots in a polynomial.
    2. Use the rational root theorem to create a list of possible rational roots.
    3. Use Descartes’ rule of signs to determine the number of positive and negative roots. Remember that multiplicity applies.
    4. Use synthetic division to try to find roots.
    5. While using synthetic division, pay attention to the sign changes of the results. Use the upper bound and lower bound
    6. theorems to eliminate higher or lower values from the list of possible roots.
    7. Factor the polynomial using synthetic division, factoring rules, or the quadratic formula. Remember that the conjugate radical theorem and complex conjugate theorem apply.
    8. Identify the roots that are solutions.

Alg2B 5.5 Example

 


Examples

  • Ex 1. What are the roots of the polynomial below:
    • f(x)=x^3-x^2-9x+9
      • Step 1: list the possible roots: p = 9, q = 1
        • \frac{p}{q}: Since q = 1, you won’t have a denominator: 9, -9, 3, -3
      • Step 2: try each root using synthetic division
        • If the remainder is zero, you have a root!
      • Answer: The roots are 1, 3, and -3.
        • Notice that when the root is 1, the resulting polynomial quotient is: x^2-9.
        • This is factored into (x-3)(x+3), so you know that 3 and -3 are roots without having to do all of this synthetic division!

Alg2B 5.5 Ex1a Alg2B 5.5 Ex1b

 

 

  • Ex 2. What are all the possible rational roots of the polynomial below:
    • f(x)=2x^3+5x^2-8x-10=0
    • List the possible roots: p = 10, q = 2
      • \frac{p}{q}: \frac{\pm 1, \pm 2, \pm 5, \pm 10}{\pm 1, \pm 2}
      • Simplified: \pm 1, \pm \frac{1}{2}, \pm 2, \pm 5, \pm \frac{5}{2}, \pm 10

 

  • Ex 3. Which of the following expresses the possible number of positive real solutions for the polynomial equation shown below?
    • 5x^3+x^2+7x-28=0
    • Answer: one

 

  • Ex 4. Which of the following expresses the possible number of positive real solutions for the polynomial equation shown below?
    • 5x^3+x^2-7x-28=0
    • Answer: Two or Zero

 

  • Ex 5. A polynomial has one root that equals 4 + 17i. What is the other root of this polynomial?
    • Answer: 4 – 17i

 

  • Ex 6. How many solutions over the complex number system does this polynomial have?
    • 3x^6-x^3-4x^2+3x+52=0
    • Answer: 6

 

  • Ex 7. Express the polynomial as a product of linear factors.
    • f(x)=x^3-5x^2-18x+72
    • Find all possible roots, then try them using sythetic division. Once you find one, factor the remaining quotient to find the rest!
    • Answer: (x-3)(x+4)(x-6)

 

  • Ex 8. What is the sum of the roots of the polynomial shown below?
    • f(x)=3x^3+12x^2+3x-18
    • Find all of the roots, then combine them.
    • Answer: -3-2+1=4

 

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