How to Factor Polynomials with Coefficients

Factoring polynomials with coefficients is a common problem in precalculus courses.
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A polynomial is a mathematical expression that consists of variables and coefficients constructed together using basic arithmetic operations, such as multiplication and addition. An example of a polynomial is the expression x^3 - 20x^2 + 100x. The process of factoring a polynomial means simplifying a polynomial into the simplest form that makes the statement true. The problem of factoring polynomials frequently arises in precalculus courses, but performing this operation with coefficients can be completed in a few short steps.

    Remove any common factors from the polynomial, if possible. As an example, the terms in the polynomial x^3 - 20x^2 + 100x have the common factor 'x'. Therefore, the polynomial can be simplified to x(x^2 - 20x + 100).

    Determine the form of the terms that remain to be factored. In the example above, the term x^2 - 20x + 100 is a quadratic with a leading coefficient of 1 (that is, the number in front of the highest power variable, which is x^2, is 1), and therefore can be solved using a specific method to solve problems of this type.

    Factor the remaining terms. The polynomial x^2 - 20x + 100 can be factored into the form x^2 + (a+b)x + ab, which can also be written as (x - a)(x - b), where 'a' and 'b' are numbers that are to be determined. Therefore, the factors are found by determining two numbers 'a' and 'b' that add up to -20 and equal 100 when multiplied together. Two such numbers are -10 and -10. The factored form of this polynomial is then (x - 10)(x - 10), or (x - 10)^2.

    Write the fully factored form of the full polynomial, including all terms that were factored. Concluding the example above, the polynomial x^3 - 20x^2 + 100x was first factored by factoring 'x', giving x(x^2 - 20x +100), and factoring the polynomial within the brackets gives x(x - 10)^2, which is the fully factored form of the polynomial.

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