How to Solve Binomial Equations by Factoring

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Instead of solving x^4 + 2x^3 = 0, factoring the binomial means you solve two simpler equations: x^3 = 0 and x + 2 = 0. A binomial is any polynomial with two terms; the variable can have any whole-number exponent of 1 or higher. Learn which binomial forms to solve by factoring. In general, they are those you can factor down to an exponent of 3 or less. Binomials can have multiple variables, but you can seldom solve those with more than one variable by factoring.

  • Check whether the equation is factorable. You can factor a binomial that has a greatest common factor, is a difference of squares, or is a sum or difference of cubes. Equations such as x + 5 = 0 can be solved without factoring. Sums of squares, such as x^2 + 25 = 0, are not factorable.

  • Simplify the equation and write it in standard form. Move all the terms to the same side of the equation, add like terms and order the terms from highest to lowest exponent. For example, 2 + x^3 - 18 = -x^3 becomes 2x^3 -16 = 0.

  • Factor out the greatest common factor, if there is one. The GCF may be a constant, a variable or a combination. For example, the greatest common factor of 5x^2 + 10x = 0 is 5x. Factor it to 5x(x + 2) = 0. You could not factor this equation any further, but if one of the terms is still factorable, as in 2x^3 - 16 = 2(x^3 - 8), continue the factoring process.

  • Use the appropriate equation to factor a difference of squares or a difference or sum of cubes. For a difference of squares, x^2 - a^2 = (x + a)(x - a). For example, x^2 - 9 = (x + 3)(x - 3). For a difference of cubes, x^3 - a^3 = (x - a)(x^2 + ax + a^2). For example, x^3 - 8 = (x - 2)(x^2 + 2x + 4). For a sum of cubes, x^3 + a^3 = (x + a)(x^2 - ax + a^2).

  • Set the equation equal to zero for each set of parentheses in the fully-factored binomial. For 2x^3 - 16 = 0, for example, the fully factored form is 2(x - 2)(x^2 + 2x + 4) = 0. Set each individual equation equal to zero to get x - 2 = 0 and x^2 + 2x + 4 = 0.

  • Solve each equation to get a solution to the binomial. For x^2 - 9 = 0, for example, x - 3 = 0 and x + 3 = 0. Solve each equation to get x = 3, -3. If one of the equations is a trinomial, such as x^2 + 2x + 4 = 0, solve it using the quadratic formula, which will result in two solutions (Resource).

Tips & Warnings

  • Check your solutions by plugging each one into the original binomial. If each calculation results in zero, the solution is correct.
  • The total number of solutions should equal the highest exponent in the binomial: one solution for x, two solutions for x^2, or three solutions for x^3.
  • Some binomials have repeat solutions. For example, the equation x^4 + 2x^3 = x^3(x + 2) has four solutions, but three are x = 0. In such cases, record the repeating solution only once; write the solution for this equation as x = 0, -2.

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