Final answer:
The zeros of the polynomial -3(2x + 3x⁵ - x)(5x - 6) are x = 0 and x = 6/5. Options (b), (c), and (d) are not zeros of the polynomial.
Step-by-step explanation:
To find the zeros of the polynomial f(x) = -3(2x + 3x⁵ - x)(5x - 6), set the polynomial equal to zero and solve for x.
First, distribute the -3:
f(x) = -3(2x + 3x⁵ - x)(5x - 6) = -3(2x×(5x - 6) + 3x⁵×(5x - 6) - x×(5x - 6))
The polynomial simplifies to:
-3((10x² - 12x) + (15x⁶ - 18x⁵) - (5x² - 6x))
However, factoring or expanding this does not give a simple polynomial required to apply the quadratic formula. Instead, notice that the zeros of f(x) are where any of the factors equal zero:
2x + 3x⁵ - x = 0 or 5x - 6 = 0
These factors can be simplified to:
x(2 + 3x⁴ - 1) = 0 and 5x - 6 = 0
The solutions for x where the expression equals zero are:
- x = 0 (because x times anything equals zero, making the first factor equal to zero)
- x = 6/5 (from 5x - 6 = 0, solving for x we get x = 6/5)
For the part x(2 + 3x⁴ - 1) being equal to zero, apart from the factored x, there are no real roots because the polynomial inside the parentheses does not factor into linear terms with real roots.
Therefore, the zeros of the polynomial are x = 0 and x = 6/5. None of the options (b), (c), or (d) are zeros of this polynomial.