Edexcel PURE 2024 October — Question 3

Exam BoardEdexcel
ModulePURE
Year2024
SessionOctober
PaperDownload PDF ↗
TopicFactor & Remainder Theorem
TypeTwo unknowns with show-that step
DifficultyModerate -0.8 This is a straightforward application of the Remainder and Factor Theorems with routine algebraic manipulation. Part (a) requires substituting x=-3 into f(x) and setting equal to 55, part (b) uses f(5/2)=0 to solve simultaneous equations, and part (c) is simple polynomial division or substitution. All steps are standard textbook exercises requiring only direct application of learned techniques with no problem-solving insight needed.
Spec1.02j Manipulate polynomials: expanding, factorising, division, factor theorem
3. $$f ( x ) = 2 x ^ { 3 } - x ^ { 2 } + A x + B$$ where \(A\) and \(B\) are integers.
Given that when \(\mathrm { f } ( x )\) is divided by \(( x + 3 )\) the remainder is 55
  1. show that $$3 A - B = - 118$$ Given also that \(( 2 x - 5 )\) is a factor of \(\mathrm { f } ( x )\),
  2. find the value of \(A\) and the value of \(B\).
  3. Hence find the quotient when \(\mathrm { f } ( x )\) is divided by ( \(x - 7\) )
Question 3:
AnswerMarks
3(a)2 (−3 )3−(−3 )2 + A(−3 )+B=55
or e.g.
AnswerMarks
− 5 4 − 9 − 3 A + B = 5 5M1
− 5 4 − 9 − 3 A + B = 5 5
AnswerMarks
 3 A − B = − 1 1 8 *A1*
(2)
AnswerMarks Guidance
2x2−7x 21 + A
x2x3 −7x2
36x2 −21x
(b)3 2
 5   5   5 
2 − + A + B = 0
AnswerMarks
2 2 2M1
3A−B=−118, 5A+2B=−50
AnswerMarks
 A = . .. , o r B = . ..M1
A = – 26, B = 40A1
(3)
AnswerMarks Guidance
x22x A
5 +
2
AnswerMarks Guidance
2x2x3 4x2
−5−5x2 − 1 0 x
− 2 5 −
2
AnswerMarks Guidance
(c)f ( x ) = ( x − 7 ) ( 2 x 2 + . .. x + . .. ) + . .. M1
2 x 2 + 1 3 x + 6 5A1
(2)

Total 7

AnswerMarks Guidance
2x21 3 x 6 5
x2x3 13x2
−7−14x2 − 9 1 x 
Question 3:
--- 3(a) ---
3(a) | 2 (−3 )3−(−3 )2 + A(−3 )+B=55
or e.g.
− 5 4 − 9 − 3 A + B = 5 5 | M1
− 5 4 − 9 − 3 A + B = 5 5
 3 A − B = − 1 1 8 * | A1*
(2)
2x2 | −7x | 21 + A
x | 2x3 | −7x2 | (21 + A)x
3 | 6x2 | −21x | 63 + 3A
(b) | 3 2
 5   5   5 
2 − + A + B = 0
2 2 2 | M1
3A−B=−118, 5A+2B=−50
 A = . .. , o r B = . .. | M1
A = – 26, B = 40 | A1
(3)
x2 | 2x | A
5 +
2
2x | 2x3 | 4x2 | ( 1 0 + A ) x
−5 | −5x2 | − 1 0 x | 5 A
− 2 5 −
2
(c) | f ( x ) = ( x − 7 ) ( 2 x 2 + . .. x + . .. ) + . .. | M1
2 x 2 + 1 3 x + 6 5 | A1
(2)
Total 7
2x2 | 1 3 x | 6 5
x | 2x3 | 13x2 | 6 5 x
−7 | −14x2 | − 9 1 x | − 4 5 5
3.

$$f ( x ) = 2 x ^ { 3 } - x ^ { 2 } + A x + B$$

where $A$ and $B$ are integers.\\
Given that when $\mathrm { f } ( x )$ is divided by $( x + 3 )$ the remainder is 55
\begin{enumerate}[label=(\alph*)]
\item show that

$$3 A - B = - 118$$

Given also that $( 2 x - 5 )$ is a factor of $\mathrm { f } ( x )$,
\item find the value of $A$ and the value of $B$.
\item Hence find the quotient when $\mathrm { f } ( x )$ is divided by ( $x - 7$ )
\end{enumerate}

\hfill \mbox{\textit{Edexcel PURE 2024 Q3}}
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