Question

complex numbers online practice
complete this assessment to review what you’ve learned. it will not count to
which of the following functions has three roots? (1 point)
y = 3x⁸ + 9x⁶ − 5x + 12
y = x⁴ − x² + 15
y = 6x⁶ + 8x³ − x + 3
y = 8x³ + x² − 2x + 1

Explanation:

Step1: Recall the Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that a polynomial of degree \( n \) has exactly \( n \) roots (counting multiplicities and complex roots). So we need to find the degree of each polynomial.

Step2: Find the degree of each function

• For \( y = 3x^{8}+9x^{6}-5x + 12 \): The highest power of \( x \) is 8, so degree is 8.
• For \( y = x^{4}-x^{2}+15 \): The highest power of \( x \) is 4, so degree is 4.
• For \( y = 6x^{6}+8x^{3}-x + 3 \): The highest power of \( x \) is 6, so degree is 6.
• For \( y = 8x^{3}+x^{2}-2x + 1 \): The highest power of \( x \) is 3, so degree is 3.

A polynomial of degree 3 will have 3 roots (by the Fundamental Theorem of Algebra, considering multiplicities and complex roots). So the function \( y = 8x^{3}+x^{2}-2x + 1 \) has three roots.

Answer:

\( y = 8x^{3}+x^{2}-2x + 1 \) (the last option)