Problem

For \(y=3(x-1)(x-4)\), explain why the nonzero factor \(3\) does not create or move a zero. Solve the two zero equations, state the \(x\text{-}\text{intercepts}\) and multiplicities, and predict the graph’s behavior at each.

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The nonzero scale factor 3 does not change the zeros. The zeros are 1 and 4, so the x-intercepts are (1, 0) and (4, 0); each has multiplicity 1, so the graph crosses at both.

The factor equations give zeros −1 and −4, so the intercepts are (−1, 0) and (−4, 0). Solving gives 1 and 4 instead.

Using 3 = 0 as a zero equation gives x = 3 as an intercept. A nonzero constant factor never equals zero.

The zeros are 1 and 4, with multiplicity 1 each, so the graph crosses at both. The required x-intercepts are not written as ordered pairs.

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For \(y=3(x-1)(x-4)\), explain why the nonzero factor \(3\) does not create or move a zero. Solve the two zero equations, state the \(x\text{-}\text{intercepts}\) and multiplicities, and predict the graph’s behavior at each.

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