How to Solve Polynomial Equations by Factoring

Written by johnmcgee
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A polynomial equation is a sum of variable terms, such as x^2+3x+1=0, where "^2" means x is squared. Factoring a polynomial into products of binomials, x-s, solving a polynomial equation then reduces to inspection. For example, C(x-s1)(x-s2)(x-s3)=0 has the solutions s1, s2 and s3, for constant C. How to factor a polynomial P(x) to solve the formula P(x)=0 can range from trial and error to complex algorithms. Sometimes, the best strategy is to guess one of the solutions so that you can reduce the order of the polynomial and make it easier to factor more fully.

Skill level:
Moderate

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Instructions

    Second-order Trinomial: The Bridge Method

  1. 1

    Factor P(x) when in the form axe^2+bx+c=0 by first moving the "a" to the "c," and multiplying them both. For example, 2x^2+2x-4=0 becomes x^2+2x-8=0.

  2. 2

    Determine two numbers that multiply to the last constant and sum to the constant of x. For example, 4 and -2 multiply to give -8 and add to give 2.

  3. 3

    Use the numbers found in Step 2 to factor the quadratic. For example, x^2+2x-8=0 becomes (x+4)(x-2)=0.

  4. 4

    Divide by constants by the original "a" and multiply the whole equation by the original "a." For example, (x+4)(x-2)=0 becomes 2(x+2)(x-1)=0. So now you know by inspection that -2 and 1 solve the original equation P(x)=0.

    Cubics: Special Cases

  1. 1

    Factor the form x^3-y^3 as (x-y)(x^2+xy+y^2). For example, 27a^3-64 = (3x-4)((3x)^2+3x*4+4^2). Then of course 3x-4=0 will give you one solution to P(x)=0. You can use the bridge method or the quadratic formula to factor the quadratic to get the other solutions.

  2. 2

    Factor the form x^3+y^3 as (x+y)(x^2-xy+y^2). You can use the bridge method or the quadratic formula to factor the quadratic to get the other solutions.

  3. 3

    Factor the form x^3+px-q = 0, where p and q are constants, by replacing x with w-p/(3w). The result is a quadratic equation in w^3, which you can then factor using the bridge method or the quadratic formula. Solving for w, you'd then solve for x using x=w-p/3w to solve the original equation.

    Higher Order: Guess and Test

  1. 1

    Take some guesses as to a solution, s, of the equation P(x)=0. For example, you may guess that 3 is a solution of x^3-7x-33=0. You may use numerical (iterative) methods to make the guess, for example using the bisection method. Once you do find such a solution, s, proceed to the next step to reduce the order of P(x) to make factoring for other solutions easier.

  2. 2

    Divide the monomial x-s into the polynomial formula, P(x). It will divide in evenly, without a remainder, since s is a zero of P(x). Denote the polynomial resulting from the division H(x). So P(x) = (x-s)H(x).

  3. 3

    Factor H(x) using the quadratic formula, the bridge method, special-case cubic equations, or repeat Step 1 to reduce its order further.

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