How to Factor Quartic Polynomials

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Quartic polynomials are polynomials of degree four. These polynomials can have as many as four real factors, factors not involving complex numbers, and as few as none. Graphing the equations is the quick way to tell how many factors to expect. A graph can also give you an idea how many of the factors are real and how many are complex. The graph can also help you see which factor candidates to try first.

Things You'll Need

  • Graphing calculator
  • Graph the equation. Each time the graphed curve crosses the x-axis represents a real monomial factor. The place where the curve crosses the x-axis is a root of the equation and x - p, where p is the point where the curve crosses the axis -- is a real valued monomial factor. Complex roots always come in pairs, so the number of real valued monomial factors will be 0, 2 or 4. You cannot really get the factors from the graph, even if there are four of them, but the graph does indicate the type of factors to expect.

  • Look at the first and last numbers in the quartic equation to find candidates for the polynomial factors. For example, for 2X^4 -13X^3 +28X^2 -23X + 6 the first number is 2, which has factors 1 and 2. The last number is 6 which has factors 1, 2, 3 and 6. The candidates for factors for the quartic are X - 1, X + 1, X - 2, X + 2, X - 3, X + 3, X - 6, X + 6, 2X - 1, 2X + 1, 2X - 2, 2X + 2, 2X - 3, 2X + 3,2 X - 6 and 2X + 6. Trying each of these, we find that 2X^4 -13X^3 +28X^2 -23X + 6 = (X - 1)(X - 2)(X - 3)(2X - 1). This quartic has four real roots. If two of the roots were complex, we would have found two monomial divisors. If all of the roots were complex, none of the candidates would be divisors.

  • Factor the binomial factors using the quadratic formula. For some applications, complex roots are undesirable, so the binomial factors are left un-factored. For example, 4X^4 - X^3 - 2X^2 - 2X + 4 = (X - 1)(X - 2)(X^2 + 2X +2). None of the other monomial candidates divide 4X^4 - X^3 - 2X^2 - 2X + 4. You can use the quadratic formula to factor X^2 + 2X +2 into complex monomials: X^2 + 2X +2 = (X + 1 + i)(X + 1 - i), so 4X^4 - X^3 - 2X^2 - 2X + 4 = (X - 1)(X - 2)(X + 1 + i)(X + 1 - i). The application determines if factoring all the way to complex numbers is required.

Tips & Warnings

  • The Quadratic formula states that AZ^2 + BZ + C has roots Z = (-B + (B^2 - 4AC)^0.5)/2A and Z = (-B - (B^2 - 4AC)^0.5)/2A.
  • You can not rely on the graph alone for finding roots of high-order equations. For example, if one root is (X - p)^3, the graph of the curve will go through point p only one time. This is true for any odd multiple root.

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