The factorized form of the given function, f(x) = 2x⁴ - x³ - 18x² + 9x, is, f(x) = x(2x - 1)(x + 3)(x - 3), and the complete statement is, From left to right, function f has zeros at x = -3, x = 0, x = 1/2, and x = 3.
In the question, we are asked to find the factors of the function f, and complete the given statement.
The function f, given to us is:
f(x) = 2x⁴ - x³ - 18x² + 9x.
To factor the function, we first group the terms as follows:
f(x) = (2x⁴ - 18x²) - (x³ - 9x).
Now, we take common terms out of the group as follows:
f(x) = 2x²(x² - 9) - x(x² - 9).
Now, we again take common terms, as follows:
f(x) = (2x² - x)(x² - 9),
or, f(x) = x(2x - 1)(x + 3)(x - 3).
This is the factored form of the given function, f.
The statement given to us is:
From left to right, function f has zeros at x = _, x = _, x = _, and x = _.
To find the zeroes of the function, we equate the factorized form to 0.
x(2x - 1)(x + 3)(x - 3) = 0.
By zero-product rule, the zeroes are at:
x = 0,
2x - 1 = 0, or, x = 1/2,
x + 3 = 0, or, x = -3,
x - 3 = , or, x = 3.
Arranging these in ascending order, the zeroes are at: -3, 0, 1/2, and 3.
Thus, the factorized form of the given function, f(x) = 2x⁴ - x³ - 18x² + 9x, is, f(x) = x(2x - 1)(x + 3)(x - 3), and the complete statement is, From left to right, function f has zeros at x = -3, x = 0, x = 1/2, and x = 3.
Learn more about factorizing a function at
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For the complete question, refer to the attachment.
