To determine which function has three roots, we can analyze the degree of each polynomial and potentially find its roots.
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y = x^4 − x^2 + 15: This is a 4th degree polynomial. A polynomial of degree n can have at most n roots. However, since the polynomial is positive for all real values of x (the minimum value occurs at the vertex, which would provide a positive value), it has no real roots.
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y = 6x^6 + 8x^3 − x + 3: This is a 6th degree polynomial, which can have up to 6 roots. Without specific calculations, we generally cannot confirm the number of their real roots.
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y = 3x^8 + 9x^6 − 5x + 12: This is an 8th degree polynomial, which can have up to 8 roots. Again, without more information, we cannot ascertain the specific number of roots.
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y = 8x^3 + x^2 − 2x + 1: This is a 3rd degree polynomial. A cubic polynomial can have either 1 or 3 real roots. To check this, we can find its derivative and analyze the behavior of the function:
- Derivative: \(y' = 24x^2 + 2x - 2\)
- Determine the critical points by setting \(y' = 0\):
\[ 24x^2 + 2x - 2 = 0 \]
Using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-2 \pm \sqrt{(2)^2 - 4 \cdot 24 \cdot (-2)}}{2 \cdot 24} = \frac{-2 \pm \sqrt{4 + 192}}{48} = \frac{-2 \pm 14}{48} \] This gives two critical points:
- \(x_1 = \frac{12}{48} = \frac{1}{4}\)
- \(x_2 = \frac{-16}{48} = -\frac{1}{3}\)
Now we evaluate the function at a few values to check the behavior:
- \(y(-1) = 8(-1)^3 + (-1)^2 - 2(-1) + 1 = -8 + 1 + 2 + 1 = -4\)
- \(y(0) = 8(0)^3 + (0)^2 - 2(0) + 1 = 1\)
- \(y(1) = 8(1)^3 + (1)^2 - 2(1) + 1 = 8 + 1 - 2 + 1 = 8\)
Since \(y(-1) < 0\) and \(y(0) > 0\), and we know there are turning points, we can conclude that this cubic function crosses the x-axis three times. Therefore, it has three real roots.
Thus, the function that has three roots is:
y = 8x^3 + x^2 − 2x + 1.
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