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Chapter 2: Problem 101

Find all real zeros of the function. $$f(y)=4 y^{3}+3 y^{2}+8 y+6$$

Short Answer

Expert verified
The function \(f(y)=4 y^{3}+3 y^{2}+8 y+6 = 0\) has no real roots.

Step by step solution

01

Equating the Function to Zero

The first step is to set the function equal to zero. This gives: \(4y^{3}+3y^{2}+8y+6 = 0\)
02

Searching for Rational Roots

From using the Rational Root Theorem, try to find the possible rational roots of the equation. The rational roots of an equation are all fractions \(\frac{p}{q}\) where \(p\) is a factor of the constant (in this case, +6) and \(q\) is a factor of the leading coefficient (in this case, +4). By checking all possible \(\frac{p}{q}\), it is found that none of these rational numbers are roots of the given polynomial, meaning the polynomial has no rational roots.
03

Using the Cubic Formula

Since there are no rational roots, use the cubic formula (which is out of scope for high school level) to solve for the roots. However, it is known that the roots of this cubic equation are all complex numbers by the Fundamental Theorem of Algebra.

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Most popular questions from this chapter

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