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Solving polynomial equations

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Solving Polynomial Equations

A polynomial equation sets a polynomial equal to zero, and solving it means finding the roots. Learn the factoring method step by step: set the equation to zero, factor, apply the zero product property to split it, and solve each factor. Includes worked quadratic and higher-degree examples.

What a polynomial equation is

A polynomial equation sets a polynomial equal to zero, such as x² − 5x + 6 = 0. Solving it means finding every value of the variable that makes the equation true — these values are called the roots or solutions. The degree of the polynomial tells you how many roots to expect.

The most common method is factoring, which relies on the zero product property: if a product equals zero, then at least one of its factors must equal zero. The steps are always the same.

Solving a polynomial equation by factoring Four steps applied to x squared minus 5x plus 6 equals 0. Step 1 set the equation equal to zero. Step 2 factor into (x minus 2)(x minus 3) equals 0. Step 3 set each factor equal to zero. Step 4 solve to get x equals 2 or x equals 3. 1 Set equal to zerox² − 5x + 6 = 0 2 Factor(x − 2)(x − 3) = 0 3 Set each factor = 0x − 2 = 0 or x − 3 = 0 4 Solve eachx = 2 or x = 3
Solving a polynomial equation by factoring: set to zero, factor, split, and solve.

Factoring often starts by pulling out a common factor, so it helps to be quick at taking out the greatest common factor. Many polynomial equations are quadratics in disguise, which you can handle the same way you solve quadratic equations by factoring.

Take x² − 5x + 6 = 0. It is already equal to zero, so factor the left side into (x − 2)(x − 3) = 0. By the zero product property, x − 2 = 0 or x − 3 = 0, which gives x = 2 or x = 3. Both values are roots — substitute them back in to check.

A cubic or higher polynomial is solved the same way once it is fully factored. For example, x³ − x = 0 factors to x(x − 1)(x + 1) = 0, giving three roots: x = 0, x = 1, and x = −1. These ideas carry into word problems of polynomials, where the equation models a real situation.

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