Moderate -0.3 This is a straightforward multi-part question requiring standard techniques: verifying a factor (routine substitution), polynomial division/factorization (standard A-level skill), and a simple substitution (e^x = y) to solve the exponential equation. While part (c) adds a minor twist, the entire question follows predictable patterns with no novel problem-solving required, making it slightly easier than average.
4. Let \(f ( x )\) be given by:
$$f ( x ) = x ^ { 3 } + x ^ { 2 } - 12 x - 18$$
a) Use the factor theorem to show that ( \(x + 3\) ) is a factor of \(f ( x )\)
b) Factorise \(f ( x )\) into a linear and a quadratic factor and hence find exact values for all of the solutions of the equation \(f ( x ) = 0\), showing detailed reasoning with your working
c) Hence write down the one solution to the equation
$$e ^ { 3 x } + e ^ { 2 x } - 12 e ^ { x } - 18 = 0$$
in the form \(\ln ( a + \sqrt { b } )\)
[0pt]
4. Let $f ( x )$ be given by:
$$f ( x ) = x ^ { 3 } + x ^ { 2 } - 12 x - 18$$
a) Use the factor theorem to show that ( $x + 3$ ) is a factor of $f ( x )$\\
b) Factorise $f ( x )$ into a linear and a quadratic factor and hence find exact values for all of the solutions of the equation $f ( x ) = 0$, showing detailed reasoning with your working\\
c) Hence write down the one solution to the equation
$$e ^ { 3 x } + e ^ { 2 x } - 12 e ^ { x } - 18 = 0$$
in the form $\ln ( a + \sqrt { b } )$\\[0pt]
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\hfill \mbox{\textit{SPS SPS FM 2022 Q4 [6]}}