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Given the polynomial y = 10x³ – 27x? + 15x – 2, a. list all possible numerators for rational roots of this polynomial: b. list all possible denominators for rational roots of this polynomial: c. list all possible rational roots:

Given the polynomial y = 10x³ – 27x? + 15x – 2, a. list all possible numerators for rational roots of this polynomial: b. list all possible denominators for rational roots of this polynomial: c. list all possible rational roots:

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Given the polynomial \( y = 10x^3 - 27x^2 + 15x - 2 \), a. List all possible numerators for rational roots of this polynomial: [ ] b. List all possible denominators for rational roots of this polynomial: [ ] c. List all possible rational roots: [ ]
Transcribed Image Text:Given the polynomial \( y = 10x^3 - 27x^2 + 15x - 2 \), a. List all possible numerators for rational roots of this polynomial: [ ] b. List all possible denominators for rational roots of this polynomial: [ ] c. List all possible rational roots: [ ]
**Given the polynomial \( y = 29x - 10x^2 - 21 \):** - **List the possible rational roots:** To find the possible rational roots of a polynomial, you can use the Rational Root Theorem. This theorem states that any rational solution, expressed as \(\frac{p}{q}\), will have \(p\) as a factor of the constant term and \(q\) as a factor of the leading coefficient. For the polynomial \( y = -10x^2 + 29x - 21 \): - The constant term is \(-21\). - The leading coefficient is \(-10\). **Factors of the constant term (\(-21\))**: \[ \pm 1, \pm 3, \pm 7, \pm 21 \] **Factors of the leading coefficient (\(-10\))**: \[ \pm 1, \pm 2, \pm 5, \pm 10 \] **Possible rational roots** (by the Rational Root Theorem) are combinations of \(\frac{p}{q}\): \[ \pm 1, \pm \frac{1}{2}, \pm \frac{1}{5}, \pm \frac{1}{10}, \pm 3, \pm \frac{3}{2}, \pm \frac{3}{5}, \pm \frac{3}{10}, \pm 7, \pm \frac{7}{2}, \pm \frac{7}{5}, \pm \frac{7}{10}, \pm 21, \pm \frac{21}{2}, \pm \frac{21}{5}, \pm \frac{21}{10} \]
Transcribed Image Text:**Given the polynomial \( y = 29x - 10x^2 - 21 \):** - **List the possible rational roots:** To find the possible rational roots of a polynomial, you can use the Rational Root Theorem. This theorem states that any rational solution, expressed as \(\frac{p}{q}\), will have \(p\) as a factor of the constant term and \(q\) as a factor of the leading coefficient. For the polynomial \( y = -10x^2 + 29x - 21 \): - The constant term is \(-21\). - The leading coefficient is \(-10\). **Factors of the constant term (\(-21\))**: \[ \pm 1, \pm 3, \pm 7, \pm 21 \] **Factors of the leading coefficient (\(-10\))**: \[ \pm 1, \pm 2, \pm 5, \pm 10 \] **Possible rational roots** (by the Rational Root Theorem) are combinations of \(\frac{p}{q}\): \[ \pm 1, \pm \frac{1}{2}, \pm \frac{1}{5}, \pm \frac{1}{10}, \pm 3, \pm \frac{3}{2}, \pm \frac{3}{5}, \pm \frac{3}{10}, \pm 7, \pm \frac{7}{2}, \pm \frac{7}{5}, \pm \frac{7}{10}, \pm 21, \pm \frac{21}{2}, \pm \frac{21}{5}, \pm \frac{21}{10} \]
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