Which statement about the polynomial function [tex]g(x)[/tex] is true?

A. If all rational roots of [tex]g(x)=0[/tex] are integers, the leading coefficient of [tex]g(x)[/tex] must be 1.

B. If all roots of [tex]g(x)=0[/tex] are integers, the leading coefficient of [tex]g(x)[/tex] must be 1.

C. If the leading coefficient of [tex]g(x)[/tex] is 1, all rational roots of [tex]g(x)=0[/tex] must be integers.
D. If the leading coefficient of [tex]g(x)[/tex] is 1, all roots of [tex]g(x)=0[/tex] must be integers.

The true statement about the polynomial function g ( x ) is that if the leading coefficient is 1, all rational roots of g ( x ) = 0 must be integers. Therefore, the correct answer is option C.
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Answered by Anonymous | 2025-07-04

Statement 1 is false because a polynomial like 2 x − 2 has an integer root but a leading coefficient of 2.
Statement 2 is false for the same reason as statement 1; 2 x − 2 provides a counterexample.
Statement 3 is true because, by the Rational Root Theorem, if the leading coefficient is 1, any rational root must be an integer.
Statement 4 is false because a polynomial like x 2 − x + 1 has a leading coefficient of 1 but non-integer roots.
Therefore, the correct answer is that if the leading coefficient of g ( x ) is 1, all rational roots of g ( x ) = 0 must be integers. I f t h e l e a d in g coe ff i c i e n t o f g ( x ) i s 1 , a ll r a t i o na l roo t so f g ( x ) = 0 m u s t b e in t e g ers . ​

Explanation

Analyzing the Statements Let's analyze each statement about the polynomial function g ( x ) to determine which one is true. We'll use examples and the Rational Root Theorem to evaluate each statement.

Evaluating Statement 1 Statement 1: If all rational roots of g ( x ) = 0 are integers, the leading coefficient of g ( x ) must be 1.


Consider the polynomial g ( x ) = 2 x − 2 . The rational root is x = 1 , which is an integer. However, the leading coefficient is 2, not 1. Therefore, statement 1 is false.

Evaluating Statement 2 Statement 2: If all roots of g ( x ) = 0 are integers, the leading coefficient of g ( x ) must be 1.

Consider the polynomial g ( x ) = 2 x − 2 . The root is x = 1 , which is an integer. However, the leading coefficient is 2, not 1. Therefore, statement 2 is false.

Evaluating Statement 3 Statement 3: If the leading coefficient of g ( x ) is 1, all rational roots of g ( x ) = 0 must be integers.

Let g ( x ) = x n + a n − 1 ​ x n − 1 + ⋯ + a 1 ​ x + a 0 ​ be a polynomial with a leading coefficient of 1. By the Rational Root Theorem, any rational root p / q of g ( x ) must have p dividing a 0 ​ and q dividing the leading coefficient, which is 1. Thus, q must be 1, so p / q = p /1 = p , which is an integer. Therefore, statement 3 is true.

Evaluating Statement 4 Statement 4: If the leading coefficient of g ( x ) is 1, all roots of g ( x ) = 0 must be integers.

Consider the polynomial g ( x ) = x 2 − x + 1 . The leading coefficient is 1, but the roots are given by the quadratic formula: x = 2 ( 1 ) − ( − 1 ) ± ( − 1 ) 2 − 4 ( 1 ) ( 1 ) ​ ​ = 2 1 ± 1 − 4 ​ ​ = 2 1 ± i 3 ​ ​ These roots are not integers. Therefore, statement 4 is false.

Conclusion Based on our analysis, statement 3 is the only true statement.

Examples
Understanding the relationship between the leading coefficient and the roots of a polynomial is crucial in various fields, such as engineering and physics. For example, when designing a control system, engineers need to determine the stability of the system, which depends on the roots of a characteristic polynomial. If the leading coefficient is 1, and we know that all rational roots must be integers, it simplifies the analysis and design process, allowing for more efficient and reliable systems.

Answered by GinnyAnswer | 2025-07-04