Which of the following represents the zeros of f(x) = 4x³ - 13x² - 37x + 10?
5, -2, -1/4
5, -2, 1/4
5, 2, -1/4
5, 2, 1/4

Solution:

Using the Rational Zeros Theorem, which states that, if the polynomial f(x) = a_nxⁿ + a_{n - 1}x^{n - 1} + ... + a₁x + a₀ has integer coefficients,

then every rational zero of f(x) has the form p/q where p is a factor of the constant term a₀ and q is a factor of the leading coefficient a_n.

Given:

Function f(x) = 4x³ - 13x² - 37x + 10

Here,

p: ±1, ±2, ±5, ±10 which are all factors of constant term 10.

q: ±1, ±2, ±4 which are all factors of the leading coefficient 4

All possible values are

p/q: ±1, ±2, ±5, ±10, ±1/2, ±1/4, ±5/2, ±5/4

From the given options we can select only ±5/4, ±2/4, ±1/4 to verify the roots.

f(x) = 4x³ - 13x² - 37x + 10

f(5/4) = 4(5/4)³ - 13(5/4)² - 37(5/4) + 10 = -48.75

f(5/4) = 4(-5/4)³ - 13(-5/4)² - 37(-5/4) + 10 = 28.125

f(2/4) = 4(2/4)³ - 13(2/4)² - 37(2/4) + 10 = -11.25

f(-2/4) = 4(-2/4)³ - 13(-2/4)² - 37(-2/4) + 10 = 24.75

f(1/4) = 4(1/4)³ - 13(1/4)² - 37(1/4) + 10 = 0

The given options are not the zeros of the given polynomial.

However on verification it is identified that 5, -2 and 1/4.

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Which of the following represents the zeros of f(x) = 4x³ - 13x² - 37x + 10?

Summary:

The zeros of the f(x) = 4x³ - 13x² - 37x + 10 are 5, -2, 1/4. We have used rational root theorem to find the zeros.