Moderate -0.8 This is a straightforward application of the Remainder Theorem requiring students to substitute two values into the polynomial and solve simultaneous linear equations. The arithmetic involves fractions but is routine, and the method is a standard textbook exercise with no problem-solving insight required—easier than average.
3 The polynomial \(2 x ^ { 3 } + a x ^ { 2 } + b x + 6\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). When \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\) the remainder is - 38 and when \(\mathrm { p } ( x )\) is divided by \(( 2 x - 1 )\) the remainder is \(\frac { 19 } { 2 }\).
Find the values of \(a\) and \(b\).
Substitute \(x = \frac{1}{2}\) and equate the result to \(\frac{19}{2}\), or divide by \(2x-1\) to obtain quadratic quotient, and equate constant remainder to \(\frac{19}{2}\).
Obtain a correct evaluated equation, e.g. \(\frac{1}{4} + \frac{a}{4} + \frac{b}{2} + 6 = \frac{19}{2}\) or \(\frac{a}{4} + \frac{b}{2} = \frac{13}{4}\)
A1
Obtain \(a = -3\) and \(b = 8\)
A1
ISW
5
## Question 3:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $2(-2)^3 + a(-2)^2 + b(-2) + 6 = -38$ | M1 | Substitute $x = -2$ and equate the result to $-38$, or divide by $x+2$ to obtain quadratic quotient, and equate constant remainder to $-38$. |
| Obtain a correct evaluated equation, e.g. $-16 + 4a - 2b + 6 = -38$ or $4a - 2b = -28$ | A1 | |
| $2\left(\frac{1}{2}\right)^3 + a\left(\frac{1}{2}\right)^2 + b\left(\frac{1}{2}\right) + 6 = \frac{19}{2}$ | M1 | Substitute $x = \frac{1}{2}$ and equate the result to $\frac{19}{2}$, or divide by $2x-1$ to obtain quadratic quotient, and equate constant remainder to $\frac{19}{2}$. |
| Obtain a correct evaluated equation, e.g. $\frac{1}{4} + \frac{a}{4} + \frac{b}{2} + 6 = \frac{19}{2}$ or $\frac{a}{4} + \frac{b}{2} = \frac{13}{4}$ | A1 | |
| Obtain $a = -3$ and $b = 8$ | A1 | ISW |
| | **5** | |
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3 The polynomial $2 x ^ { 3 } + a x ^ { 2 } + b x + 6$, where $a$ and $b$ are constants, is denoted by $\mathrm { p } ( x )$. When $\mathrm { p } ( x )$ is divided by $( x + 2 )$ the remainder is - 38 and when $\mathrm { p } ( x )$ is divided by $( 2 x - 1 )$ the remainder is $\frac { 19 } { 2 }$.
Find the values of $a$ and $b$.\\
\hfill \mbox{\textit{CAIE P3 2023 Q3 [5]}}