Determine zeros from factored form

Factored form exposes a polynomial’s real zeros: by the zero-product property, a product equals zero exactly when at least one factor equals zero, so each linear factor is set equal to zero; a nonzero constant factor contributes no zero. Repeated factors identify zeros with multiplicity, and the real zeros correspond to the function’s $x$ -intercepts, supporting connections between algebraic form and graphs; non-real zeros and more advanced root theory are not included.

Detailed Explanation: Determine zeros from factored form

When a polynomial is written in factored form, its zeros can be found by using the zero-product property:

A product equals zero when at least one of its factors equals zero.

A nonzero constant, such as $-3$ , does not create a zero. Set each linear factor equal to zero and solve.

Example

Find the real zeros of

f(x)=-3(x-2)^2(x+5).

Step 1: Identify the factors.

The factors are

-3,\qquad (x-2)^2,\qquad (x+5).

The constant factor $-3$ is nonzero, so it does not affect where the function equals zero.

Step 2: Set each linear factor equal to zero.

For the factor $x-2$ :

x-2=0

x=2

For the factor $x+5$ :

x+5=0

x=-5

Step 3: State the zeros and their multiplicities.

The zeros are

\boxed{x=2\text{ and }x=-5}.

Because $(x-2)$ is squared, $x=2$ has multiplicity $2$ . The factor $(x+5)$ appears once, so $x=-5$ has multiplicity $1$ .

The corresponding $x$ -intercepts of the graph are

\boxed{(2,0)\text{ and }(-5,0)}.

Learn by doing: Determine zeros from factored form

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Polynomial Inequalities - Two Factors with Multiplicity - Sign Chart

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