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

To determine the zeroes and their multiplicities for the function \( f(x) = 3x^6 + 30x^5 + 75x^4 \), let's factor the equation.

1. Factor out the greatest common factor (GCF):
   \[
   f(x) = 3x^4(x^2 + 10x + 25)
   \]
   This simplifies the equation, leaving \( 3x^4 \) and a quadratic expression.

2. Factor the quadratic:
   \[
   x^2 + 10x + 25 = (x + 5)^2
   \]
   Therefore, the fully factored form is:
   \[
   f(x) = 3x^4(x + 5)^2
   \]

Now, examine the factored form \( 3x^4(x + 5)^2 \):

- The factor \( x^4 \) gives a zero at \( x = 0 \) with **multiplicity 4**.
- The factor \( (x + 5)^2 \) gives a zero at \( x = -5 \) with **multiplicity 2**.

Thus, the correct description of the zeroes is:

\[
\boxed{-5 \text{ with multiplicity 2 and } 0 \text{ with multiplicity 4}}
\]

Would you like more details or have any questions about this?

Here are 5 related questions:
1. What does the multiplicity of a zero tell you about the graph at that point?
2. How does the graph behave near zeroes with even and odd multiplicities?
3. Can you factor other polynomials using the same method as in this example?
4. How does the degree of a polynomial affect the number of zeroes it can have?
5. How do changes in coefficients affect the zeroes of a polynomial?

**Tip**: Zeroes with higher multiplicity cause the graph to "flatten" or touch the x-axis without crossing it.

Which of the following describes the zeroes of the graph of f(x) = 3x^6 + 30x^5 + 75x^4?
–5 with multiplicity 2 and 0 with multiplicity 4
5 with multiplicity 2 and 0 with multiplicity 4
–5 with multiplicity 2 and 0 with multiplicity 4
5 with multiplicity 2 and 0 with multiplicity 4

Discover how to find the zeroes and their multiplicities for the polynomial function f(x) = 3x^6 + 30x^5 + 75x^4. This step-by-step solution involves factoring and analyzing the behavior of zeroes, including a zero at x = 0 with multiplicity 4 and a zero at x = -5 with multiplicity 2. Suitable for students in Grades 10-12.

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