Question

27. Which one of the following is TRUE about the graph of the polynomial function f(x) = x(x-1)²(x + 1)³? A. It touches the x-axis at x=0 C. It crosses the x-axis at x=1 B. It touches the x-axis at x=1 D. It touches the x-axis at x=-1 ​

Answer 1

Answer:

The graph of this function touches the $x$-axis at $x = 1$.

Step-by-step explanation:

The function in this question is expressed as the product of its linear factors: $x$, $(x - 1)$, and $(x + 1)$. By the Factor Theorem, the graph of this function will either cross or touch the $x$-axis at each point where any of these factors becomes zero:

• $x = 0$ for the factor $x$,
• $x = 1$ for the factor $(x - 1)$, and
• $x = (-1)$ for the factor $(x + 1)$ (which is equivalent to $(x - (-1))$.)

Whether the graph crosses or touches the $x$-axis at each of these point depends on the multiplicity of the factor.

Specifically, for a linear factor of the form $(x - x_{0})$, the function would cross $x$-axis at $x = x_{0}$ if the multiplicity of this factor is an odd number. Otherwise, the graph of the function would touch the $x$-axis without crossing at $x = x_{0}\!$.

• The factor $x$ is raised to a power of $1$. The multiplicity of this factor is $1\!$. The graph of this function would cross $x$-axis at $x = 0$.
• The factor $(x - 1)$ is raised to a power of $2$. The multiplicity of this factor is $2\!$. The graph of this function would touch $x$-axis at $x = 1$.
• The factor $(x + 1)$ is raised to a power of $3$. The multiplicity of this factor is $3$. The graph of this function would cross $x$-axis at $x = (-1)$.

Hence, among the choices, only the statement that the graph of this function touches the $x$-axis at $x = 1$ is valid.