Step-by-step explanation:
To identify the zeros, we need to factorize the expression as much as possible.
So, let's try dividing the function by (3x + 1). If the remainder is zero (3x + 1) is one of the factors of the polynomial.
Then, f(x) = 6x⁴ + 35x³ + 11x² - 300x - 100 divided by (3x + 1) is equal to:
Therefore, (3x + 1) is a factor and the other factor is (2x³ + 11x² - 100).
Now, let's try with (2x - 5), so (2x³ + 11x² - 100)² divided by (2x - 5) is equal to:
Therefore, (2x - 5) is another factor of f(x)
Now, we can rewrite the expression as:
f(x) = (3x + 1)(2x - 5)(x² + 8x + 20)
So, the zeros of the polynomial are:
3x + 1 = 0
3x + 1 - 1 = 0 - 1
3x = -1
3x/3 = -1/3
x = -1/3
2x - 5 = 0
2x - 5 + 5 = 0 + 5
2x = 5
2x/2 = 5/2
x = 5/2
And the solutions for the equation (x² + 8x + 20) = 0 can be calculated by the quadratic equation as:
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