# How to Factor Using Substitution

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Factoring is a process used to find the elements that go into a polynomial. A polynomial is a math expression that includes variables raised to powers. The expression x^2 + 5x + 6 is a polynomial. This expression is an example of a simple trinomial (three terms) that can be factored into two terms, (x+2) (x+3). These terms are considered factors because when they are multiplied together, they result in the original expression. When faced with a more complicated polynomial, substitution can be a handy method for finding the factors.

## Using Substitution

• When faced with a complicated trinomial such as x^6 + 8x^3 -20, the first step is to simplify the expression into a common quadratic (highest power of 2) trinomial. To do so, you take the middle term, in this case, 8x^3, and substitute another variable for x^3. In this case, use a=x^3 as your substitution.

• Rewrite the expression using your substituted variable. Since a=x^3, the expression is rewritten as a^2+8a-20. It is now in standard quadratic trinomial form and is ready for factoring.

• Follow the standard steps in factoring polynomials to determine what products result in -20 as a final term and 8 as a middle term. In this case, 10 and -2 fit the bill. Thus, the factored expression is (a+10)(a-2).

• Substitute x^3 for a to get back to the original terms. Thus, (a+10) (a-2) becomes (x^3 + 10)(x^3-2).

• Check your answer by multiplying the terms out. If they result in the original expression, you have factored it correctly.

## Tips & Warnings

• Sometimes the factors themselves will be factorable. Check to make sure that your factors are in the simplest terms before completing the problem.

## References

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