Show that x = -2 is a root of the polynomial equation 15x³ + 26x² - 11x − 6 = 0.

Mark scheme

Marking points from the official mark scheme (May/June 2015). The mark or mark type is shown in brackets.

1. [B1] -120 + 104 + 22 - 6 = 0 or correct unsimplified form, e.g. 15(-2)³ + 26(-2)² – 11(-2) – 6 = 0 or 15(-8) + 26(4) – 11(-2) – 6 = 0 or correct synthetic division -2 | 15 26 -11 -6 | -30 8 6 | 15 -4 -3 0
2. [M1] Substituting x = 3 into 15x³ + 26x² – 11x – 6 or correct synthetic division 3 | 15 26 -11 -6 | 45 213 606 | 15 71 202 600
3. [A1] 600 correct answer implies M1; must be explicitly identified as answer if using synthetic/long division methods by e.g. circling
4. [B1] (x - 1)(15x³ + 26x² – 11x – 6) soi by inspection or division; may be implied by e.g. (ax + b)(15x³ + 26x² – 11x – 6) and a = 1, b = -1 seen in later work comparing coefficients
5. [M1] Multiply out (x + 1)(15x³ + 26x² – 11x – 6) and compare coefficients of x³ or x to quartic or multiply out, e.g. (ax + b)(15x³ + 26x² – 11x – 6) and compare coefficients of x³ or x to quartic
6. [A1] p = 11 correct p or q implies M1; correct p and q www implies B1 M1
7. [A1] q = 5