Consider the following function.
f(x) = x5 + x3 − 42x
(a)
Find all real zeros of the polynomial function. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.)
x =    

We are given the function \( f(x) = x^5 + x^3 - 42x \). To find all real zeros, we need to solve for \( x \) such that \( f(x) = 0 \).

### Step 1: Factor the polynomial
First, we can factor out the common factor of \( x \):
\[
f(x) = x(x^4 + x^2 - 42) = 0
\]
This gives us two cases to solve:
1. \( x = 0 \)
2. \( x^4 + x^2 - 42 = 0 \)

### Step 2: Solve \( x^4 + x^2 - 42 = 0 \)
Let’s make a substitution: let \( u = x^2 \). This simplifies the equation to:
\[
u^2 + u - 42 = 0
\]
This is a quadratic equation. We can solve it using the quadratic formula:
\[
u = \frac{-1 \pm \sqrt{1^2 - 4(1)(-42)}}{2(1)}
\]
\[
u = \frac{-1 \pm \sqrt{1 + 168}}{2}
\]
\[
u = \frac{-1 \pm \sqrt{169}}{2}
\]
\[
u = \frac{-1 \pm 13}{2}
\]
Thus, the solutions for \( u \) are:
\[
u = \frac{-1 + 13}{2} = 6 \quad \text{or} \quad u = \frac{-1 - 13}{2} = -7
\]

### Step 3: Solve for \( x \)
Recall that \( u = x^2 \), so we have two cases:
1. \( x^2 = 6 \) gives \( x = \pm \sqrt{6} \)
2. \( x^2 = -7 \) has no real solutions since the square of a real number cannot be negative.

### Step 4: List all real solutions
The real solutions are:
\[
x = 0, \pm \sqrt{6}
\]

Thus, the real zeros of the function are:
\[
x = 0, \sqrt{6}, -\sqrt{6}
\]

Would you like any more details or further clarifications?

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Here are 5 related questions you might find helpful:
1. How can you verify if a function is factored correctly?
2. What is the quadratic formula and how is it applied?
3. How do you handle equations involving higher powers like \( x^5 \)?
4. What are imaginary solutions, and when do they appear in polynomials?
5. What are some methods to approximate irrational solutions like \( \sqrt{6} \)?

**Tip**: Always check for common factors first when factoring polynomials!

Consider the following function. f(x) = x^5 + x^3 − 42x (a) Find all real zeros of the polynomial function. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.) x =

Explore how to find the real zeros of the polynomial function f(x) = x^5 + x^3 − 42x. This step-by-step solution covers factoring polynomials and applying the quadratic formula to solve higher-degree equations. Suitable for high school students in Grades 10-12.

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