Find all rational zeros of each function. $\text{ƒ}\left(x\right)=8x^4-14x^3-29x^2-4x+3$

Verified step by step guidance

1. Identify the polynomial function: \(f(x) = 8x^4 - 14x^3 - 29x^2 - 4x + 3\).
2. List all possible rational zeros using the Rational Root Theorem. The possible rational zeros are of the form \(\pm \frac{p}{q}\), where \(p\) divides the constant term (3) and \(q\) divides the leading coefficient (8). So, possible values of \(p\) are \(\pm1, \pm3\) and possible values of \(q\) are \(\pm1, \pm2, \pm4, \pm8\).
3. Form the complete list of possible rational zeros: \(\pm1, \pm\frac{1}{2}, \pm\frac{1}{4}, \pm\frac{1}{8}, \pm3, \pm\frac{3}{2}, \pm\frac{3}{4}, \pm\frac{3}{8}\).
4. Test each possible rational zero by substituting it into \(f(x)\) to check if it equals zero. This can be done by direct substitution or synthetic division.
5. Once a zero is found, use polynomial division (synthetic or long division) to divide \(f(x)\) by the corresponding factor \((x - r)\), where \(r\) is the zero found, to reduce the polynomial degree and repeat the process to find all rational zeros.

Key Concepts

Rational Root Theorem

The Rational Root Theorem helps identify possible rational zeros of a polynomial by considering factors of the constant term and the leading coefficient. For ƒ(x), possible rational roots are ±(factors of 3) divided by ±(factors of 8). This narrows down candidates to test.

Polynomial Division (Synthetic or Long Division)

Polynomial division is used to divide the original polynomial by a candidate root's corresponding factor (x - r). If the remainder is zero, r is a root, and the quotient is a reduced polynomial to further factor or solve.

Factoring and Solving Polynomial Equations

After identifying a root and dividing, the resulting polynomial can be factored further or solved using methods like factoring quadratics or applying the quadratic formula. This process helps find all rational zeros of the original polynomial.