Which of the following functions has three roots? (1 point)

Question

Responses

y=x4−x2+15
𝑦
=
𝑥
4
−
𝑥
2
+
15
y equals x superscript 4 baseline minus x squared plus 15

y=8x3+x2−2x+1
𝑦
=
8
𝑥
3
+
𝑥
2
−
2
𝑥
+
1
y equals 8 x cubed plus x squared minus 2 x plus 1

y=6x6+8x3−x+3
𝑦
=
6
𝑥
6
+
8
𝑥
3
−
𝑥
+
3
y equals 6 x superscript 6 baseline plus 8 x cubed minus x plus 3

y=3x8+9x6−5x+12

Answer 1

To determine which of the given functions has three roots, we need to analyze the degree of each polynomial and consider the potential for roots based on the Fundamental Theorem of Algebra, which states that a polynomial of degree \( n \) can have up to \( n \) roots.

Let's analyze each function:

1. \( y = x^4 - x^2 + 15 \)

   • Degree: 4
   • Can have 0, 1, 2, 3, or 4 roots.
   • Since the leading coefficient is positive and the constant term is positive, this polynomial does not have any real roots.

2. \( y = 8x^3 + x^2 - 2x + 1 \)

   • Degree: 3
   • Can have 1, 2, or 3 roots (counting multiplicities).
   • Since this is a cubic function, it will always have at least one real root, and based on the nature of cubic functions, it can have two additional real roots or fewer.

3. \( y = 6x^6 + 8x^3 - x + 3 \)

   • Degree: 6
   • Can have 0, 1, 2, 3, 4, 5, or 6 roots.
   • This does not guarantee three roots.

4. \( y = 3x^8 + 9x^6 - 5x + 12 \)

   • Degree: 8
   • Can have 0 to 8 roots.
   • Again, this does not guarantee three roots.

Conclusion:

The only polynomial that can specifically have three roots is:

\( y = 8x^3 + x^2 - 2x + 1 \)

This is a cubic function, and while it could potentially have either 1 or 3 real roots (depending on its discriminant), it is the only polynomial in the list with a degree that permits exactly three roots.