Mathematics 2019 JAMB Past Questions - Question 16: Find the value of p and q such that (x-1) and ( x-3) are factors of px3+ qx2 + 11x-6

Correct Answer

B

Explanation

To find the values of \( p \) and \( q \) such that \( (x-1) \) and \( (x-3) \) are factors of \( px^3 + qx^2 + 11x - 6 \), we can use the factor theorem.According to the factor theorem, if \( (x-a) \) is a factor of a polynomial, then the polynomial will be equal to zero when \( x = a \).So, for \( (x-1) \) to be a factor, \( p(1)^3 + q(1)^2 + 11(1) - 6 = 0 \), which simplifies to \( p + q + 11 - 6 = 0 \) or \( p + q + 5 = 0 \) ... (Equation 1)Similarly, for \( (x-3) \) to be a factor, \( p(3)^3 + q(3)^2 + 11(3) - 6 = 0 \), which simplifies to \( 27p + 9q + 33 - 6 = 0 \) or \( 27p + 9q + 27 = 0 \) or \( 3p + q + 3 = 0 \) ... (Equation 2)We now have a system of linear equations:\( p + q + 5 = 0 \) ... (Equation 1)\( 3p + q + 3 = 0 \) ... (Equation 2)Solving these equations simultaneously will give us the values of \( p \) and \( q \).