Problem 29
Question
In Exercises \(25-32,\) 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. $$f(x)=x^{3}-2 x^{2}+x$$
Step-by-Step Solution
Verified Answer
The zeros of the function \(f(x)=x^{3}-2 x^{2}+x\) are \(x=0\) with multiplicity 1 and \(x=1\) with multiplicity 2. At \(x=0\), the graph crosses the x-axis and at \(x=1\), the graph touches the x-axis and turns around.
1Step 1: Factoring the Polynomial
Start by factoring the given polynomial function: \(f(x)=x^{3}-2 x^{2}+x\). Factor out the greatest common factor which is \(x\) : \[x(x^{2}-2x+1)\]
2Step 2: Solve for Zeroes
Set each factor equal to zero and solve for \(x\): \(x=0\) and \(x^{2}-2x+1=0\). The quadratic factor can be factored further to \( (x-1)^2 = 0\), where the roots are \(x=0\) and \(x=1\)
3Step 3: Determine Multiplicity and Graph Behavior
The root \(x=0\) occurs only once, so its multiplicity is 1, which means the graph will cross the x-axis at \(x=0\). The root \(x=1\) occurs twice, so its multiplicity is 2. When the multiplicity is even, the graph touches the x-axis and turns around at \(x = 1\).
Key Concepts
Polynomial Factorization
Polynomial Factorization
Understanding the process of polynomial factorization is essential to finding the zeros of a polynomial function. This fundamental algebraic method breaks down a complex polynomial into a product of simpler polynomials or factors. These simpler factors are often easier to solve when looking for roots (or zeros) of the original polynomial.
For example, consider the polynomial function given by
For example, consider the polynomial function given by
Other exercises in this chapter
Problem 28
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