Answer :
To find all the zeros of the polynomial function [tex]\( f(x) = 4x^4 + 16x^3 + 23x^2 + 14x + 3 \)[/tex], we can follow these steps:
1. Identify the polynomial: We are given [tex]\( f(x) = 4x^4 + 16x^3 + 23x^2 + 14x + 3 \)[/tex]. This is a quartic polynomial, meaning its highest degree is 4. We are looking for values of [tex]\( x \)[/tex] which make [tex]\( f(x) = 0 \)[/tex].
2. Try rational root theorem (optional): The Rational Root Theorem can help identify possible rational roots. According to this theorem, any rational solution, written as a fraction p/q, where p and q are integers, is such that p is a factor of the constant term (here, 3) and q is a factor of the leading coefficient (here, 4).
3. Verify potential rational roots: You would test potential rational roots such as [tex]\(\pm 1, \pm \frac{1}{2}, \pm 3, \pm \frac{3}{2}, \pm 2, \pm \frac{3}{4}, \pm \frac{1}{4}\)[/tex], etc., by substituting them into [tex]\( f(x) \)[/tex] and checking if they satisfy [tex]\( f(x) = 0 \)[/tex].
4. Factorization and simplifying roots: Upon determining the rational roots from the calculations, we factorize the polynomial based on these roots. Factoring the polynomial step by step or using a synthetic division can help simplify it. After confirming factors like [tex]\((x + 1)\)[/tex] or [tex]\((x + \frac{3}{2})\)[/tex] divide the polynomial, continue until you simplify fully to identify more roots.
5. Find the remaining zeros: The roots identified through this process are the zeros of the polynomial: [tex]\(-\frac{3}{2}, -1, -\frac{1}{2}\)[/tex].
Therefore, the zeros of the polynomial [tex]\( f(x) = 4x^4 + 16x^3 + 23x^2 + 14x + 3 \)[/tex] are [tex]\( x = -\frac{3}{2}, -1, -\frac{1}{2} \)[/tex].
1. Identify the polynomial: We are given [tex]\( f(x) = 4x^4 + 16x^3 + 23x^2 + 14x + 3 \)[/tex]. This is a quartic polynomial, meaning its highest degree is 4. We are looking for values of [tex]\( x \)[/tex] which make [tex]\( f(x) = 0 \)[/tex].
2. Try rational root theorem (optional): The Rational Root Theorem can help identify possible rational roots. According to this theorem, any rational solution, written as a fraction p/q, where p and q are integers, is such that p is a factor of the constant term (here, 3) and q is a factor of the leading coefficient (here, 4).
3. Verify potential rational roots: You would test potential rational roots such as [tex]\(\pm 1, \pm \frac{1}{2}, \pm 3, \pm \frac{3}{2}, \pm 2, \pm \frac{3}{4}, \pm \frac{1}{4}\)[/tex], etc., by substituting them into [tex]\( f(x) \)[/tex] and checking if they satisfy [tex]\( f(x) = 0 \)[/tex].
4. Factorization and simplifying roots: Upon determining the rational roots from the calculations, we factorize the polynomial based on these roots. Factoring the polynomial step by step or using a synthetic division can help simplify it. After confirming factors like [tex]\((x + 1)\)[/tex] or [tex]\((x + \frac{3}{2})\)[/tex] divide the polynomial, continue until you simplify fully to identify more roots.
5. Find the remaining zeros: The roots identified through this process are the zeros of the polynomial: [tex]\(-\frac{3}{2}, -1, -\frac{1}{2}\)[/tex].
Therefore, the zeros of the polynomial [tex]\( f(x) = 4x^4 + 16x^3 + 23x^2 + 14x + 3 \)[/tex] are [tex]\( x = -\frac{3}{2}, -1, -\frac{1}{2} \)[/tex].
