Solving Polynomial Equations by Factoring

From a Polynomial Equation to Its Solutions

A polynomial equation can look complicated because its terms are spread across both sides, or because its factors are hidden. A useful strategy is to make the equation equal to zero, factor the resulting polynomial, and use the fact that a product is zero only when at least one of its factors is zero.

The order matters. The zero-product property applies to a product equal to zero, not to a product equal to an arbitrary number. Once the equation is correctly arranged and factored, the factors reveal the possible solutions. Checking them in the original equation confirms that they work.

Why Set the Equation Equal to Zero?

Suppose an equation is written as \(P(x)=Q(x)\), where \(P\) and \(Q\) are polynomials. Subtracting \(Q(x)\) from both sides gives an equivalent equation, \(P(x)-Q(x)=0\). The solutions have not changed: a number makes the original sides equal exactly when their difference is zero.

After collecting like terms, factor the polynomial on the left as far as possible. Earlier material on factoring covers methods such as taking out a greatest common factor, factoring quadratic trinomials, using special product identities, grouping, and substitution. Here the goal is to use a suitable factorization to solve the equation.

The Zero-Product Property

This property explains why factoring can turn one equation into several simpler equations. For example, if a polynomial equation becomes \((x-2)(x+5)=0\), then one of the two factors must be zero. Solving \(x-2=0\) and \(x+5=0\) produces the possible values of \(x\).

Why the Zero-Product Property Works

For two factors, suppose \(ab=0\). If \(a=0\), the first factor is zero. If \(a\ne 0\), then \(a\) has a multiplicative inverse \(1/a\). Multiplying both sides of \(ab=0\) by \(1/a\) gives \(b=0\). Thus \(a=0\) or \(b=0\).

The result extends to any finite number of factors. If \(a_1a_2\cdots a_n=0\), apply the two-factor result to \(a_1\) and \(a_2\cdots a_n\). Either \(a_1=0\), or the remaining product is zero. Applying the same reasoning to that remaining product shows that some factor must be zero. This is why each factor in a factored polynomial equation supplies a possible equation to solve.

Worked Examples

A Reliable Solving Process

When Factoring Does Not Produce Only Linear Factors

Factoring can reveal all real solutions when the resulting factors can be solved over the real numbers. Sometimes, however, a factor remains that has no real zeros. For example, \(x^2+1\) is positive for every real \(x\), since \(x^2\ge 0\) and therefore \(x^2+1\ge 1\). Thus \(x^2+1=0\) has no real solution.

This distinction matters when checking that a solution list is complete. If a polynomial has been factored into linear factors, setting each factor to zero identifies every real solution. If a remaining factor is not linear, determine whether it has real solutions rather than simply ignoring it. A factorization method may also fail to expose useful factors; that does not by itself prove that the equation has no solutions.

A constant polynomial gives two special cases. If the equation reduces to a nonzero constant equal to zero, it has no solutions. If it reduces to \(0=0\), every real number satisfies it. In the usual factoring problems, the polynomial is nonconstant, but recognizing these cases prevents treating an empty or zero expression as an ordinary product of variable factors.

Common Mistake: Using the Property Before the Product Equals Zero

The zero-product property does not say that a factor of a product equal to \(4\), \(7\), or any other nonzero number must be zero. For example, from \(x(x-3)=4\), it would be incorrect to claim \(x=0\) or \(x-3=0\). The right-hand side is not zero, so the property does not apply.

Instead, move all terms to one side and factor the resulting zero equation:

Now the zero-product property applies, giving \(x=4\) or \(x=-1\). Check the original equation: for \(x=4\), \(4(4-3)=4\); for \(x=-1\), \((-1)(-1-3)=4\). Both are valid. The correction is not merely to factor; it is to make sure the equation is equal to zero before applying the property.

Why the Method Matters

Factoring is useful because it converts one polynomial equation into simpler equations whose solutions can be read from their factors. The zero-product property provides the logical justification that the list is complete: any solution makes the entire product zero, so at least one factor must vanish. Conversely, a value that makes any factor zero makes the whole product zero.

For that reasoning to give a reliable answer, keep three checks in view: the original equation must be rearranged correctly, the factorization must multiply back to the polynomial, and each proposed value must satisfy the original equation. These checks catch sign errors, missed factors, and mistakes from applying the zero-product property too early.