Which of the following describes the zeroes of the graph of *f*(*x*) = 3*x*6 + 30*x*5 + 75*x*4?


–5 with multiplicity 2 and  with multiplicity 4

5 with multiplicity 2 and  with multiplicity 4

–5 with multiplicity 2 and 0 with multiplicity 4

5 with multiplicity 2 and 0 with multiplicity 4

Let's find the zeroes of the function \( f(x) = 3x^6 + 30x^5 + 75x^4 \).

### Step 1: Factor the function
First, factor out the greatest common factor (GCF) from the terms:
\[
f(x) = 3x^4(x^2 + 10x + 25)
\]

### Step 2: Factor the quadratic expression
Next, factor the quadratic expression \( x^2 + 10x + 25 \). Notice that it can be factored as a perfect square:
\[
x^2 + 10x + 25 = (x + 5)^2
\]

### Step 3: Write the fully factored form of the function
Substituting the factored quadratic into the original function:
\[
f(x) = 3x^4(x + 5)^2
\]

### Step 4: Determine the zeroes and their multiplicities
- The factor \( x^4 \) indicates a zero at \( x = 0 \) with multiplicity 4.
- The factor \( (x + 5)^2 \) indicates a zero at \( x = -5 \) with multiplicity 2.

### Conclusion
The zeroes of the function are:
- \( x = 0 \) with multiplicity 4
- \( x = -5 \) with multiplicity 2

Thus, the correct description of the zeroes is:
- **–5 with multiplicity 2 and 0 with multiplicity 4**

Would you like further details or have any questions?

Here are 5 related questions:
1. How does the multiplicity of a root affect the graph of a polynomial function?
2. What is the significance of the leading coefficient in determining the end behavior of a polynomial function?
3. How can you determine the number of turning points in the graph of a polynomial function?
4. What is the relationship between the degree of a polynomial and the maximum number of roots it can have?
5. How does factoring a polynomial help in solving equations?

**Tip:** When factoring polynomials, always look for the greatest common factor (GCF) first, as it simplifies the expression and makes further factoring easier.

Explore the zeroes of the polynomial function f(x) = 3x^6 + 30x^5 + 75x^4. Learn how to factor polynomials and determine their zeroes with detailed examples and explanations. Suitable for high school students in Grades 10-12.

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