How to Split the Middle Term

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Splitting the middle term is an effective method for factoring polynomials with a leading coefficient other than 1. The guess-and-check method, which involves finding two numbers that multiply to make the constant C and add to make the coefficient B, only works for monic quadratic equations, where A = 1. In all other cases, you have to use the guess-and-check method to split the middle term into two separate terms, then factor by grouping to solve the polynomial.

  • Write the polynomial in standard form, Ax^2 + Bx + C. This may involve combining like terms and reordering term in the polynomial. For example, you would move the constant on right side of the equation 2x^2 - 8x = -8 over to the left to get the equation 2x^2 - 8x + 8 = 0.

  • Multiply the coefficients A and C (the coefficient of the x^2 term and the constant) together. In the example polynomial, you would multiply 2 by 8 to get 16.

  • Use the guess-and-check strategy to find two whole numbers whose product is the number you found in Step 2 and whose sum is equal to B (the coefficient of the x term). In the example, you would find two numbers whose product is 16 and whose sum is -8. Those two numbers are -4 and -4.

  • Split the middle term Bx into the sum of the two terms Mx and Nx, where M and N are the two numbers you found in Step 3. In the example polynomial, you would split the term -8x into -4x - 4x, resulting in the polynomial 2x^2 - 4x - 4x + 8.

  • Factor the new polynomial by grouping. Find the greatest common factor (GCF) of the first two terms and the last two terms, then rewrite the polynomial as the product of the sum of the two GCFs and the paired factor. In the example, the GCF of 2x^2 - 4x is 2x and the GCF of 4x + 8 is 4. The factored form of the polynomial is therefore 2x(x - 2) - 4(x - 2), or (2x - 4)(x - 2).

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