Which of these rational numbers does the Rational Roots Theorem say can't be a solution to 14x7 +
13x5 − 19x4 + 7x2 − 6x − 21 = 0?
A. 3/2
B. 2/7
C. -1
D. -(1/2)
To determine which of the given rational numbers cannot be a solution to the equation, we can apply the Rational Roots Theorem.
The Rational Roots Theorem states that any rational root (solution) of a polynomial equation with integer coefficients 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.
The given equation is: 14x^7 + 13x^5 − 19x^4 + 7x^2 − 6x − 21 = 0.
First, let's find the factors of the constant term (-21) and the leading coefficient (14):
Factors of -21: ±1, ±3, ±7, ±21
Factors of 14: ±1, ±2, ±7, ±14
Now, let's check each of the given rational numbers to see if they are possible solutions:
A. 3/2:
The factors of the constant term (-21) are ±1, ±3, ±7, ±21.
The factors of the leading coefficient (14) are ±1, ±2, ±7, ±14.
Both 3 and 2 are factors of the constant term and the leading coefficient, so 3/2 is a possible solution.
B. 2/7:
The factors of the constant term (-21) are ±1, ±3, ±7, ±21.
The factors of the leading coefficient (14) are ±1, ±2, ±7, ±14.
Neither 2 nor 7 is a factor of the constant term or the leading coefficient, so 2/7 is not a possible solution.
C. -1:
The factors of the constant term (-21) are ±1, ±3, ±7, ±21.
The factors of the leading coefficient (14) are ±1, ±2, ±7, ±14.
Both 1 and -1 are factors of the constant term and the leading coefficient, so -1 is a possible solution.
D. -(1/2):
To check if -(1/2) is a possible solution, we need to consider the negative factors as well.
The factors of the constant term (-21) are ±1, ±3, ±7, ±21.
The factors of the leading coefficient (14) are ±1, ±2, ±7, ±14.
Neither 1 nor 2 is a factor of the constant term or the leading coefficient, and since we need to consider negative factors as well, -(1/2) is not a possible solution.
Therefore, the rational number that the Rational Roots Theorem says cannot be a solution to the given equation is:
D. -(1/2)