Question

identifying the key features of a polynomial function
complete the statements about the key features of the graph of $f(x) = x^5 - 9x^3$.
identify the zeroes of $f(x)$ choose three correct answers.
0 with multiplicity 1
-3 with multiplicity 1
3 with multiplicity 0
3 with multiplicity 1
0 with multiplicity 3

Explanation:

Step1: Factor the polynomial

To find the zeros of \( f(x) = x^5 - 9x^3 \), we first factor the polynomial. We can factor out the greatest common factor, which is \( x^3 \):

$$ f(x) = x^3(x^2 - 9) $$

Then, we recognize that \( x^2 - 9 \) is a difference of squares, which factors further as \( (x - 3)(x + 3) \). So, the fully factored form of the polynomial is:

$$ f(x) = x^3(x - 3)(x + 3) $$

Step2: Identify the zeros and their multiplicities

• For the factor \( x^3 \), the zero is \( x = 0 \), and the exponent of this factor is 3. Therefore, the zero \( 0 \) has a multiplicity of 3.
• For the factor \( (x - 3) \), the zero is \( x = 3 \), and the exponent of this factor is 1. Therefore, the zero \( 3 \) has a multiplicity of 1.
• For the factor \( (x + 3) \) (which can be written as \( (x - (-3)) \)), the zero is \( x = -3 \), and the exponent of this factor is 1. Therefore, the zero \( -3 \) has a multiplicity of 1.

Now, let's analyze the given options:

• "0 with multiplicity 1" is incorrect because the multiplicity of 0 is 3.
• "-3 with multiplicity 1" is correct because the exponent of \( (x + 3) \) is 1.
• "3 with multiplicity 0" is incorrect because the multiplicity of 3 is 1 (not 0).
• "3 with multiplicity 1" is correct because the exponent of \( (x - 3) \) is 1.
• "0 with multiplicity 3" is correct because the exponent of \( x^3 \) is 3.

So the three correct answers are: "-3 with multiplicity 1", "3 with multiplicity 1", and "0 with multiplicity 3".

Answer:

• $-3$ with multiplicity 1
• $3$ with multiplicity 1
• $0$ with multiplicity 3