Answer :
Answer:
4) Leading coefficient = ±6
Constant term = ±5
5) Leading coefficient = ±3
Constant term = ±14
6) Leading coefficient = ±14
Constant term = ±25
Step-by-step explanation:
The Rational Root Theorem states that if a polynomial function f(x) has integer coefficients, then any rational root of f(x) must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.
[tex]\boxed{\textsf{Rational root}=\dfrac{\sf p}{\sf q}=\dfrac{\textsf{A factor of the constant term}}{\textsf{A factor of the leading coefficient}}}[/tex]
The numerators of the rational roots represent the factors of the constant term, and the denominators of the rational roots represent the factors of the leading coefficient.
Therefore, to find the constant term from a list of possible roots, determine the number to which all the numerators are factors. Similarly, to find the leading coefficient, determine the number to which all the denominators are factors.
[tex]\dotfill[/tex]
Question 4
The possible rational roots of the function are given as:
[tex]\sf \pm 1, \pm \dfrac{1}{2}, \pm \dfrac{1}{3}, \pm \dfrac{1}{6}, \pm 5, \pm \dfrac{5}{2}, \pm \dfrac{5}{3}, \pm \dfrac{5}{6}[/tex]
Therefore:
- Factors of the constant term (numerators): ±1 and ±5
- Factors of the leading coefficient (denominators): ±1, ±2, ±3 and ±6
The number to which ±1 and ±5 are factors is ±5, so this indicates that the constant term of the function must be ±5.
The number to which ±1, ±2, ±3 and ±6 are factors is ±6, so this indicates that the leading coefficient of the function is ±6.
[tex]\dotfill[/tex]
Question 5
The possible rational roots of the function are given as:
[tex]\sf \pm 1, \pm 2, \pm 7, \pm 14, \pm \dfrac{1}{3}, \pm \dfrac{2}{3}, \pm \dfrac{7}{3}, \pm \dfrac{14}{3}[/tex]
Therefore:
- Factors of the constant term (numerators): ±1, ±2, ±7 and ±14
- Factors of the leading coefficient (denominators): ±1 and ±3
The number to which ±1, ±2, ±7 and ±14 are factors is ±14, so this indicates that the constant term of the function must be ±14.
The number to which ±1 and ±3 are factors is ±3, so this indicates that the leading coefficient of the function is ±3.
[tex]\dotfill[/tex]
Question 6
The possible rational roots of the function are given as:
[tex]\sf \pm 1, \pm 5 \pm 25, \pm \dfrac{1}{7}, \pm \dfrac{5}{7}, \pm \dfrac{25}{7}, \pm \dfrac{1}{14}, \pm \dfrac{5}{14}, \pm \dfrac{25}{14}[/tex]
Therefore:
- Factors of the constant term (numerators): ±1, ±5 and ±25
- Factors of the leading coefficient (denominators): ±1, ±7 and ±14
The number to which ±1, ±5 and ±25 are factors is ±25, so this indicates that the constant term of the function must be ±25.
The number to which ±1, ±7 and ±14 are factors is ±14, so this indicates that the leading coefficient of the function is ±14.
