Standard +0.3 This is a straightforward application of the factor theorem requiring polynomial division or coefficient comparison. Students must use the given factor to find the constant 'a' and determine the other quadratic factor—a standard P3 exercise with clear methodology, slightly above average difficulty due to working with quartic polynomials and quadratic factors rather than linear ones.
3 The polynomial \(x ^ { 4 } + 4 x ^ { 2 } + x + a\) is denoted by \(\mathrm { p } ( x )\). It is given that ( \(x ^ { 2 } + x + 2\) ) is a factor of \(\mathrm { p } ( x )\).
Find the value of \(a\) and the other quadratic factor of \(p ( x )\).
Attempt to find \(a\) and/or quadratic factor by division or by inspection
M1
Obtain partial quotient or factor \(x^2 - x\)
A1
State answer \(a = 6\)
B1
State or imply the other factor is \(x^2 - x + 3\)
A1
Guidance: The M1 is earned if division has produced a partial quotient \(x^2 + bx\), or if inspection has an unknown factor \(x^2 + bx + c\) and has reached an equation in \(b\) and/or \(c\).
SR: a correct division with unresolved constant remainder can earn M1A1B0A1.
NB: successive division by a pair of incorrect linear factors, e.g. \(x-1\) and \(x+2\) or \(x+1\) and \(x+2\), can earn M1A0 or M1A1 if their product is of the form \(x^2+x+k\).
Total: 4 marks
Attempt to find $a$ and/or quadratic factor by division or by inspection | M1 |
Obtain partial quotient or factor $x^2 - x$ | A1 |
State answer $a = 6$ | B1 |
State or imply the other factor is $x^2 - x + 3$ | A1 |
**Guidance:** The M1 is earned if division has produced a partial quotient $x^2 + bx$, or if inspection has an unknown factor $x^2 + bx + c$ and has reached an equation in $b$ and/or $c$. | |
SR: a correct division with unresolved constant remainder can earn M1A1B0A1. | |
NB: successive division by a pair of incorrect linear factors, e.g. $x-1$ and $x+2$ or $x+1$ and $x+2$, can earn M1A0 or M1A1 if their product is of the form $x^2+x+k$. | | **Total: 4 marks**
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3 The polynomial $x ^ { 4 } + 4 x ^ { 2 } + x + a$ is denoted by $\mathrm { p } ( x )$. It is given that ( $x ^ { 2 } + x + 2$ ) is a factor of $\mathrm { p } ( x )$.\\
Find the value of $a$ and the other quadratic factor of $p ( x )$.
\hfill \mbox{\textit{CAIE P3 2002 Q3 [4]}}