Asked by Yeah
Zeros of Polynomials Quick Check How many zeros does the function f (x) = x (x (x - 1) (2x+4)2 4)2 have? (1 point) The degree cannot be determined from the information provided. 3, 2, 4
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Answered by
GPT-4o mini
To find the number of zeros of the polynomial function \( f(x) = x (x (x - 1) (2x + 4)^2 4)^2 \), we can analyze the factors:
1. The term \( x \) contributes **1 zero** at \( x = 0 \).
2. The term \( x(x - 1) \) contributes **1 zero** at \( x = 1 \) (the \( x \) has already been counted once).
3. The term \( (2x + 4)^2 \) (which is squared) contributes **2 zeros** since \( 2x + 4 = 0 \) gives \( x = -2 \).
4. The outer squaring of the polynomial results in **no additional zeros**, but note that it could affect the multiplicity of existing zeros.
Now, we can summarize:
- The zero at \( x = 0 \): 1 zero
- The zero at \( x = 1 \): 1 zero
- The zero at \( x = -2 \): 2 zeros, contributes 2 (with a multiplicity of 2)
Now, combine these:
- Total unique zeros: 1 (from \( x = 0 \)) + 1 (from \( x = 1 \)) + 2 (from \( x = -2 \)) = 4.
Thus, the function has **4 zeros** in total.
Hence, the correct answer is 4.
1. The term \( x \) contributes **1 zero** at \( x = 0 \).
2. The term \( x(x - 1) \) contributes **1 zero** at \( x = 1 \) (the \( x \) has already been counted once).
3. The term \( (2x + 4)^2 \) (which is squared) contributes **2 zeros** since \( 2x + 4 = 0 \) gives \( x = -2 \).
4. The outer squaring of the polynomial results in **no additional zeros**, but note that it could affect the multiplicity of existing zeros.
Now, we can summarize:
- The zero at \( x = 0 \): 1 zero
- The zero at \( x = 1 \): 1 zero
- The zero at \( x = -2 \): 2 zeros, contributes 2 (with a multiplicity of 2)
Now, combine these:
- Total unique zeros: 1 (from \( x = 0 \)) + 1 (from \( x = 1 \)) + 2 (from \( x = -2 \)) = 4.
Thus, the function has **4 zeros** in total.
Hence, the correct answer is 4.