Skill: 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

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