Binomial times linear coefficient

The polynomial multiplying the binomial expansion is linear (degree 1), requiring multiplication of the binomial expansion by a simple linear expression like (a + bx).

12 questions · Moderate -0.6

1.04a Binomial expansion: (a+b)^n for positive integer n
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CAIE P1 2024 June Q1
4 marks Moderate -0.8
1 Find the coefficient of \(x ^ { 2 }\) in the expansion of $$( 2 - 5 x ) ( 1 + 3 x ) ^ { 10 }$$
CAIE P1 2004 June Q4
6 marks Moderate -0.8
4 Find the coefficient of \(x ^ { 3 }\) in the expansion of
  1. \(( 1 + 2 x ) ^ { 6 }\),
  2. \(( 1 - 3 x ) ( 1 + 2 x ) ^ { 6 }\).
CAIE P1 2010 November Q1
4 marks Moderate -0.8
1
  1. Find the first 3 terms in the expansion, in ascending powers of \(x\), of \(\left( 1 - 2 x ^ { 2 } \right) ^ { 8 }\).
  2. Find the coefficient of \(x ^ { 4 }\) in the expansion of \(\left( 2 - x ^ { 2 } \right) \left( 1 - 2 x ^ { 2 } \right) ^ { 8 }\).
CAIE P1 2012 November Q4
5 marks Moderate -0.3
4
  1. Find the first 3 terms in the expansion of \(\left( 2 x - x ^ { 2 } \right) ^ { 6 }\) in ascending powers of \(x\).
  2. Hence find the coefficient of \(x ^ { 8 }\) in the expansion of \(( 2 + x ) \left( 2 x - x ^ { 2 } \right) ^ { 6 }\).
Edexcel P2 2024 June Q1
5 marks Moderate -0.3
  1. (a) Find the first four terms, in ascending powers of \(x\), of the binomial expansion of
$$\left( 1 - \frac { 1 } { 6 } x \right) ^ { 9 }$$ giving each term in simplest form.
(b) Hence find the coefficient of \(x ^ { 3 }\) in the expansion of $$( 10 x + 3 ) \left( 1 - \frac { 1 } { 6 } x \right) ^ { 9 }$$ giving the answer in simplest form.
Edexcel C2 2014 June Q3
7 marks Moderate -0.8
3. (a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of $$( 2 - 3 x ) ^ { 6 }$$ giving each term in its simplest form.
(b) Hence, or otherwise, find the first 3 terms, in ascending powers of \(x\), of the expansion of $$\left( 1 + \frac { x } { 2 } \right) ( 2 - 3 x ) ^ { 6 }$$
Edexcel C2 2016 June Q5
9 marks Moderate -0.3
  1. (a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of
$$( 2 - 9 x ) ^ { 4 }$$ giving each term in its simplest form. $$f ( x ) = ( 1 + k x ) ( 2 - 9 x ) ^ { 4 } , \text { where } k \text { is a constant }$$ The expansion, in ascending powers of \(x\), of \(\mathrm { f } ( x )\) up to and including the term in \(x ^ { 2 }\) is $$A - 232 x + B x ^ { 2 }$$ where \(A\) and \(B\) are constants.
(b) Write down the value of \(A\).
(c) Find the value of \(k\).
(d) Hence find the value of \(B\).
OCR C2 2011 January Q1
6 marks Moderate -0.8
1
  1. Find and simplify the first three terms, in ascending powers of \(x\), in the binomial expansion of \(( 1 + 2 x ) ^ { 7 }\).
  2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 2 - 5 x ) ( 1 + 2 x ) ^ { 7 }\).
Pre-U Pre-U 9794/1 2012 Specimen Q6
7 marks Moderate -0.3
6
  1. Find and simplify the first four terms in the expansion of \(( 1 - 2 x ) ^ { 9 }\) in ascending powers of \(x\).
  2. In the expansion of $$( 2 + a x ) ( 1 - 2 x ) ^ { 9 }$$ the coefficient of \(x ^ { 2 }\) is 66 . Find the value of \(a\).
CAIE P1 2015 June Q3
5 marks Moderate -0.8
  1. Find the coefficients of \(x^2\) and \(x^3\) in the expansion of \((2 - x)^6\). [3]
  2. Find the coefficient of \(x^3\) in the expansion of \((3x + 1)(2 - x)^6\). [2]
WJEC Unit 1 2024 June Q9
9 marks Moderate -0.3
  1. Write down the binomial expansion of \((2 - x)^6\) up to and including the term in \(x^2\). [3]
  2. Given that $$(1 + ax)(2 - x)^6 = 64 + bx + 336x^2 + \ldots,$$ find the values of the constants \(a\), \(b\). [6]
SPS SPS FM 2024 October Q3
6 marks Moderate -0.8
  1. Find and simplify the first three terms in the expansion of \((2-5x)^5\) in ascending powers of \(x\). [3]
  2. In the expansion of \((1+ax)^2(2-5x)^5\), the coefficient of \(x\) is 48. Find the value of \(a\). [3]