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 the remaining quadratic factor through algebraic manipulation. While it involves multiple steps, the method is standard and commonly practiced in P3, making it slightly easier than average.
2 The polynomial \(x ^ { 4 } + 3 x ^ { 2 } + a\), where \(a\) is a constant, 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 \(\mathrm { p } ( x )\).
EITHER: Attempt division by \(x^2 + x + 2\) reaching a partial quotient of \(x^2 + kx\)
M1
Complete the division and obtain quotient \(x^2 - x + 2\)
A1
Equate constant remainder to zero and solve for \(a\)
M1
Obtain answer \(a = 4\)
A1
OR: Calling the unknown factor \(x^2 + bx + c\), obtain an equation in \(b\) and/or \(c\), or state without working two coefficients with the correct moduli
M1
Obtain factor \(x^2 - x + 2\)
A1
Use \(a = 2c\) to find \(a\)
M1
Obtain answer \(a = 4\)
A1
[4]
**EITHER:** Attempt division by $x^2 + x + 2$ reaching a partial quotient of $x^2 + kx$ | M1 |
Complete the division and obtain quotient $x^2 - x + 2$ | A1 |
Equate constant remainder to zero and solve for $a$ | M1 |
Obtain answer $a = 4$ | A1 |
**OR:** Calling the unknown factor $x^2 + bx + c$, obtain an equation in $b$ and/or $c$, or state without working two coefficients with the correct moduli | M1 |
Obtain factor $x^2 - x + 2$ | A1 |
Use $a = 2c$ to find $a$ | M1 |
Obtain answer $a = 4$ | A1 | [4]
2 The polynomial $x ^ { 4 } + 3 x ^ { 2 } + a$, where $a$ is a constant, 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 $\mathrm { p } ( x )$.
\hfill \mbox{\textit{CAIE P3 2007 Q2 [4]}}