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Chapter 8: Problem 87

Find the zeros of the function. $$x^{3}-4 x^{2}=0$$

Short Answer

Expert verified
The zeros are \( x = 0 \) and \( x = 4 \).

Step by step solution

01

- Factor the equation

Factor out the common term, which is \(x^2\), from the function.ewline\[ x^3 - 4x^2 = x^2(x - 4) = 0 \]
02

- Set each factor equal to zero

Set each factor of the equation \(x^2(x - 4) = 0\) equal to zero.ewline\[ x^2 = 0 \quad \text{and} \quad x - 4 = 0 \]
03

- Solve for x

Solve each equation for \(x\).ewline\[ x^2 = 0 \implies x = 0 \]ewline\[ x - 4 = 0 \implies x = 4 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

factoring polynomials
When we talk about factoring polynomials, it's all about finding what can be multiplied together to get the original polynomial. In this case, we start with the function \[x^3 - 4x^2 = 0\]The first step is to identify any common factors. Here, we can see that both terms have a common factor of \(x^2\).
So, we factor \(x^2\) out:
\[x^3 - 4x^2 = x^2(x - 4) = 0\]
By factoring out the common term, we have simplified the polynomial into a product of polynomials. Factoring makes it easier to solve the polynomial equation, which leads us to the next step.
setting factors to zero
After factoring the polynomial, the next important step is setting each factor to zero. This concept is based on the Zero Product Property, which states that if the product of two factors is zero, at least one of the factors must be zero. For our factored equation: \[x^2(x - 4) = 0\]
We set each factor to zero:
  • \(x^2 = 0\)
  • \(x - 4 = 0\)
Setting each factor to zero individually allows us to find the individual solutions to the polynomial equation. This step is crucial because it breaks the problem down into simpler, solvable parts.
solving polynomial equations
Finally, we move to solving the polynomial equations we derived from setting the factors to zero. For the equations:
  • \(x^2 = 0\)
    We can solve for \(x\) by taking the square root of both sides:
    \[x = 0\]
  • \(x - 4 = 0\)
    We solve this by adding 4 to both sides:
    \[x = 4\]
Therefore, the zeros or solutions of the original polynomial equation \(x^3 - 4x^2 = 0\) are \(x = 0\) and \(x = 4\).
These solutions mean that when \(x\) is 0 or 4, the polynomial function equals zero. We have successfully found the zeros by following these steps.

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