Standard +0.3 This is a straightforward application of the remainder theorem for polynomial division. Students need to use the fact that p(x) = q(x)(x²-x+1) + 3x+2, then either substitute roots of x²-x+1 or compare coefficients. It's slightly above routine because it involves a quadratic divisor rather than linear, but the method is standard and requires only algebraic manipulation with no novel insight.
3 The polynomial \(2 x ^ { 4 } + a x ^ { 3 } + b x - 1\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). When \(\mathrm { p } ( x )\) is divided by \(x ^ { 2 } - x + 1\) the remainder is \(3 x + 2\).
Find the values of \(a\) and \(b\).
Commence division and reach partial quotient \(2x^2 + (a \pm 2)x\)
M1
Need \(2x^2 + (a \pm 2)x\); working backwards from remainder: \(2x^2 + (\ldots)x \pm 3\) M1 \(2x^2 - x - 3\) A1
Obtain correct quotient \(2x^2 + (a+2)x + a\)
A1
Allow sign error e.g. in \(b-2\)
Set *their* linear remainder equal to part of "\(3x + 2\)" and solve for \(a\) or \(b\)
M1
Remainder \(= 3x + 2 = (b-2)x - 1 - a\). Allow for just equating \(x\) term or constant term.
Obtain answer \(a = -3\)
A1
Obtain answer \(b = 5\)
A1
Alternative Method:
Answer
Marks
Guidance
Answer
Marks
Guidance
State \(2x^4 + ax^3 + 0x^2 + bx - 1 = (x^2 - x + 1)(2x^2 + Ax + B) + 3x + 2\) and form/solve equations for \(A\) or \(B\)
M1
e.g. \(0 = B - A + 2\) and \(-1 = B + 2\)
Obtain \(A = -1\), \(B = -3\)
A1
Form and solve equations for \(a\) or \(b\)
M1
e.g. \(a = A - 2\) or \(b = -B + A + 3\)
Obtain \(a = -3\)
A1
Obtain \(b = 5\)
A1
Alternative Method (Remainder Theorem):
Answer
Marks
Guidance
Answer
Marks
Guidance
Use remainder theorem with \(x = \frac{1 \pm \sqrt{-3}}{2}\) or \(x = \frac{1 \pm i\sqrt{3}}{2}\)
M1
Allow correct use with reasonable attempt at either root in exact or decimal form; \(x^2 = \frac{-1+\sqrt{-3}}{2}\), \(x^3 = -1\), \(x^4 = \frac{-1-\sqrt{-3}}{2}\)
3 The polynomial $2 x ^ { 4 } + a x ^ { 3 } + b x - 1$, where $a$ and $b$ are constants, is denoted by $\mathrm { p } ( x )$. When $\mathrm { p } ( x )$ is divided by $x ^ { 2 } - x + 1$ the remainder is $3 x + 2$.
Find the values of $a$ and $b$.\\
\hfill \mbox{\textit{CAIE P3 2023 Q3 [5]}}