Find all rational zeros of each function. ƒ(x)=8x^4-14x^3-29x^2-4x+3

Verified step by step guidance

1. Identify the possible rational zeros using the Rational Root Theorem, which states that any rational zero, \( \frac{p}{q} \), is such that \( p \) is a factor of the constant term (3) and \( q \) is a factor of the leading coefficient (8).
2. List the factors of the constant term (3): \( \pm 1, \pm 3 \).
3. List the factors of the leading coefficient (8): \( \pm 1, \pm 2, \pm 4, \pm 8 \).
4. Form all possible rational zeros by taking each factor of the constant term and dividing it by each factor of the leading coefficient: \( \pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm \frac{1}{8}, \pm 3, \pm \frac{3}{2}, \pm \frac{3}{4}, \pm \frac{3}{8} \).
5. Test each possible rational zero by substituting it into the function \( f(x) = 8x^4 - 14x^3 - 29x^2 - 4x + 3 \) to see if it results in zero, indicating that it is indeed a rational zero.

Key Concepts

Rational Root Theorem

The Rational Root Theorem provides a method for identifying possible rational roots of a polynomial equation. It states that any rational solution, expressed as a fraction p/q, must have p as a factor of the constant term and q as a factor of the leading coefficient. This theorem helps narrow down the candidates for testing potential rational zeros.

Synthetic Division

Synthetic division is a simplified form of polynomial long division that is used to divide a polynomial by a linear factor. It allows for quick evaluation of potential roots by determining if a given value is a zero of the polynomial. If the remainder is zero, the value is a root, and the polynomial can be factored further.

Polynomial Functions

Polynomial functions are mathematical expressions involving variables raised to whole number exponents, combined using addition, subtraction, and multiplication. The degree of the polynomial indicates the highest exponent, which influences the function's behavior and the number of possible roots. Understanding the structure of polynomial functions is essential for analyzing their zeros.