Question

What are the zeros of the polynomial function 8x⁴+30x³ +35x²+15x+2?

Answer 1

To find the zeros of the polynomial function, we need to find the values of x that make the function equal to zero. We can do this by factoring the polynomial.

To begin, we can check if 2 is a zero of the function by substituting it into the polynomial:

8(2)⁴ + 30(2)³ + 35(2)² + 15(2) + 2 = 128 + 240 + 140 + 30 + 2 = 540

Since the result is not equal to zero, we know that 2 is not a zero of the function.

Next, we can divide the polynomial by (x - 2) using polynomial long division:

_______________________________________
(x - 2) | 8x⁴ + 30x³ + 35x² + 15x + 2
- (8x⁴ - 16x³)
------------------
46x³ + 35x² + 15x + 2
- (46x³ - 92x²)
------------------
127x² + 15x + 2
- (127x² - 254/127x)
-------------------
269/127x + 2
- (269/127x - 538/127)
-------------------
540/127

The remainder is 540/127, which is not equal to zero.

Therefore, the polynomial function 8x⁴ + 30x³ + 35x² + 15x + 2 does not have any rational zeros.