Finding Constants from Factor or Zero Remainder Conditions

Determine unknown constants in a polynomial given that certain expressions are factors (remainder is zero), then find the resulting quotient.

2 questions · Moderate -0.6

1.02j Manipulate polynomials: expanding, factorising, division, factor theorem
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CAIE P2 2011 June Q4
5 marks Moderate -0.8
4 The polynomial \(\mathrm { f } ( x )\) is defined by $$f ( x ) = 3 x ^ { 3 } + a x ^ { 2 } + a x + a$$ where \(a\) is a constant.
  1. Given that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\), find the value of \(a\).
  2. When \(a\) has the value found in part (i), find the quotient when \(\mathrm { f } ( x )\) is divided by ( \(x + 2\) ).
CAIE P2 2011 November Q6
8 marks Moderate -0.3
6
  1. The polynomial \(x ^ { 4 } + a x ^ { 3 } - x ^ { 2 } + b x + 2\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x - 1 )\) and \(( x + 2 )\) are factors of \(\mathrm { p } ( x )\). Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, find the quotient when \(\mathrm { p } ( x )\) is divided by \(x ^ { 2 } + x - 2\).