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The possible rational roots of f(x) = 7x³ + 9x² + 3x +11 are x = becomes ±1.) (Use "+" to indicate the symbol. So "+-1"

The possible rational roots of f(x) = 7x³ + 9x² + 3x +11 are x = becomes ±1.) (Use "+" to indicate the symbol. So "+-1"

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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**Finding Possible Rational Roots of a Polynomial Function** The polynomial function to consider is: \[ f(x) = 7x^3 + 9x^2 + 3x + 11 \] To determine the possible rational roots of this polynomial, use the Rational Root Theorem. According to this theorem, any rational solution of the polynomial equation \( f(x) = 0 \), where \( f(x) \) is a polynomial with integer coefficients, is of the form \( \pm p/q \), where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient. Let's identify the factors: - Constant Term: \( 11 \) - Factors: \( \pm 1, \pm 11 \) - Leading Coefficient: \( 7 \) - Factors: \( \pm 1, \pm 7 \) Hence, the possible rational roots \( x \) could be: \[ x = \pm \frac{1}{1}, \pm \frac{1}{7}, \pm \frac{11}{1}, \pm \frac{11}{7} \] Expressing them in a simplified form: \[ x = \pm 1, \pm \frac{1}{7}, \pm 11, \pm \frac{11}{7} \] Use the characters "+-" to indicate the \( \pm \) symbol. So "+-1" becomes \( \pm 1 \). Thus, the possible rational roots of the polynomial function \( f(x) \) are: \[ x = \pm 1, \pm \frac{1}{7}, \pm 11, \pm \frac{11}{7} \]
Transcribed Image Text:**Finding Possible Rational Roots of a Polynomial Function** The polynomial function to consider is: \[ f(x) = 7x^3 + 9x^2 + 3x + 11 \] To determine the possible rational roots of this polynomial, use the Rational Root Theorem. According to this theorem, any rational solution of the polynomial equation \( f(x) = 0 \), where \( f(x) \) is a polynomial with integer coefficients, is of the form \( \pm p/q \), where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient. Let's identify the factors: - Constant Term: \( 11 \) - Factors: \( \pm 1, \pm 11 \) - Leading Coefficient: \( 7 \) - Factors: \( \pm 1, \pm 7 \) Hence, the possible rational roots \( x \) could be: \[ x = \pm \frac{1}{1}, \pm \frac{1}{7}, \pm \frac{11}{1}, \pm \frac{11}{7} \] Expressing them in a simplified form: \[ x = \pm 1, \pm \frac{1}{7}, \pm 11, \pm \frac{11}{7} \] Use the characters "+-" to indicate the \( \pm \) symbol. So "+-1" becomes \( \pm 1 \). Thus, the possible rational roots of the polynomial function \( f(x) \) are: \[ x = \pm 1, \pm \frac{1}{7}, \pm 11, \pm \frac{11}{7} \]
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