Problem 29 Find the zeros for each polynomi... [FREE SOLUTION]

Chapter 3: Problem 29

Find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the $x$ -axis, or touches the $x$ -axis and turns around, at each zero. $$f(x)=x^{3}-2 x^{2}+x$$

Short Answer

Expert verified

The zeros of the polynomial function are $x=0$ and $x=1$. The multiplicity of the zero $x=0$ is 1 and the multiplicity of the zero $x=1$ is 2. The graph crosses the $x$-axis at $x=0$ and touches the $x$-axis and turns around at $x=1$.

Step by step solution

Factor the polynomial

The given function is $f(x)=x^{3}-2 x^{2}+x$. We can factor out an $x$ from each term to get $f(x)=x(x^{2}-2 x+1)$.

Set the factored polynomial to zero and solve for $x$

Setting the factored polynomial to zero gives us: $x(x^{2}-2 x+1)=0$. Now solve this equation by setting each factor equal to zero. For a product to be zero, at least one of the factors has to be zero. Therefore, $x=0$ or $x^{2}-2 x+1=0$. The quadratic equation $x^{2}-2 x+1=0$ can be solved to get $x=1$.

Determine the multiplicity of each zero

The multiplicity of a zero is simply the power of its corresponding factor in the function's factored form. In $f(x)=x(x^{2}-2 x+1)$, we can see that $x$ has a power of 1 (so multiplicity 1) and $(x-1)$ (from $x^{2}-2x+1$) squared equals has a power of 2 (so multiplicity 2). Hence, the multiplicities of 0 and 1 are 1 and 2 respectively.

Identify whether the graph of the function crosses or touches the $x$-axis and turns around at each zero

For a zero with odd multiplicity, the graph crosses the $x$-axis at that point. For a zero with even multiplicity, the graph touches the $x$-axis and turns around. Hence, as the multiplicity of 0 (from $x=0$) is 1 which is odd, the graph crosses the $x$-axis at $x=0$. The multiplicity of 1 (from $x^2 - 2x + 1 = 0$) is 2, which is even, so the graph touches the $x$-axis at $x=1$ and turns around.

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91影视

Find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each zero. $$f(x)=x^{3}-2 x^{2}+x$$

Short Answer

Step by step solution

Factor the polynomial

Set the factored polynomial to zero and solve for \(x\)

Determine the multiplicity of each zero

Identify whether the graph of the function crosses or touches the \(x\)-axis and turns around at each zero

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