Problem

For \(y=(x-2)(x+5)\), set each factor equal to zero, state both \(x\text{-}\text{intercepts}\), and give each zero’s multiplicity. Use the multiplicities to predict whether the graph crosses or touches the \(x\text{-}\text{axis}\) at each intercept.

Big Picture

What this problem is really about

An x-intercept is a zero written as a point, so the output must first be set equal to zero. We’ll solve each visible factor, attach a y-coordinate of zero, count how often each factor occurs, and use odd or even multiplicity to predict whether the graph crosses or only touches the axis.

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The equations x − 2 = 0 and x + 5 = 0 give zeros −2 and 5. The factor equations actually give 2 and −5.

Treating the scale as a zero gives x = 1 along with the factor zeros, so (1, 0) is an intercept. There is no factor that makes x = 1 a zero.

The zeros are 2 and −5, with multiplicity 1 each, so the graph crosses at both. The required x-intercepts are not written as ordered pairs.

Set x − 2 = 0 and x + 5 = 0. The zeros are 2 and −5, so the x-intercepts are (2, 0) and (−5, 0); each has multiplicity 1, so the graph crosses at both.

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For \(y=(x-2)(x+5)\), set each factor equal to zero, state both \(x\text{-}\text{intercepts}\), and give each zero’s multiplicity. Use the multiplicities to predict whether the graph crosses or touches the \(x\text{-}\text{axis}\) at each intercept.

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