Find constants from coefficient conditions

A question is this type if and only if you must find constants using information about specific coefficients in an expansion or product, rather than factor/remainder conditions.

4 questions · Moderate -0.3

1.02j Manipulate polynomials: expanding, factorising, division, factor theorem

Sort by: Default | Easiest first | Hardest first

OCR MEI C1 Q8

5 marks Moderate -0.3

8 You are given that

the coefficient of \(x ^ { 3 }\) in the expansion of \(\left( 5 + 2 x ^ { 2 } \right) \left( x ^ { 3 } + k x + m \right)\) is 29 ,
when \(x ^ { 3 } + k x + m\) is divided by ( \(x - 3\) ), the remainder is 59 .

Find the values of \(k\) and \(m\).

OCR H240/02 2018 March Q1

6 marks Moderate -0.8

1 Part of the graph of \(y = \mathrm { f } ( x )\) is shown below, where \(\mathrm { f } ( x )\) is a cubic polynomial. \includegraphics[max width=\textwidth, alt={}, center]{6a6316e4-7b2d-4533-988a-4863d79ce668-04_681_679_475_694}

Find \(\mathrm { f } ( - 1 )\).
Write down three linear factors of \(\mathrm { f } ( x )\). It is given that \(\mathrm { f } ( x ) \equiv a x ^ { 3 } + b x ^ { 2 } + c x + d\).
Show that \(a = - 2\).
Find \(b , c\) and \(d\).

Pre-U Pre-U 9794/2 2012 Specimen Q5

5 marks Standard +0.3

5 When \(x ^ { 4 } - 4 x ^ { 3 } + 5 x ^ { 2 } + x + a\) is divided by \(x ^ { 2 } - x + 1\), the quotient is \(x ^ { 2 } + b x + 1\) and the remainder is \(c x - 3\). Find the values of the constants \(a , b\) and \(c\).

OCR MEI C1 2010 June Q6

5 marks Moderate -0.3

You are given that • the coefficient of \(x^3\) in the expansion of \((5 + 2x^2)(x^3 + kx + m)\) is 29, • when \(x^3 + kx + m\) is divided by \((x - 3)\), the remainder is 59. Find the values of \(k\) and \(m\). [5]