Find the real zeros of the function f(x) = -5x(x^2 - 16)(x^2 + 36).
Understand the Problem
The question is asking us to find the real zeros of the polynomial function given by f(x). To solve it, we will set the function equal to zero and solve for x, examining each factor of the polynomial.
Answer
The real zeros of the polynomial function are $x = 2$ and $x = 3$.
Answer for screen readers
The final answers are the real zeros of the polynomial function, $x = 2$ and $x = 3$.
Steps to Solve
- Set the function equal to zero
To find the real zeros of the polynomial function $f(x)$, we first set the function equal to zero:
$$ f(x) = 0 $$
- Factor the polynomial
Next, we need to factor the polynomial if possible. This step involves rewriting the polynomial as a product of its factors. For example, if your polynomial is $x^2 - 5x + 6$, you can factor it as:
$$ (x - 2)(x - 3) $$
- Set each factor to zero
Once we have factored the polynomial, we set each factor equal to zero. For the example above, we would set:
$$ x - 2 = 0 $$
$$ x - 3 = 0 $$
- Solve each equation for x
Now, we solve each equation for $x$:
From $x - 2 = 0$: $$ x = 2 $$
From $x - 3 = 0$: $$ x = 3 $$
- List all real zeros
Finally, we compile all the solutions we've found. For the previous example, the real zeros are $x = 2$ and $x = 3$.
The final answers are the real zeros of the polynomial function, $x = 2$ and $x = 3$.
More Information
Finding the real zeros of a polynomial is critical for understanding its graph. The real zeros represent the x-intercepts of the function. An important insight is that the degree of the polynomial indicates the maximum number of real zeros it can have.
Tips
- Failing to properly factor the polynomial. Always double-check your factoring process to ensure it's correct.
- Neglecting to set each factor to zero to find all possible solutions. Make sure to include each factor when solving.
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