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Show that \( (x + 1)(x + 2)(x + 3) \) can be written in the form \( ax^3 + bx^2 + cx + d \) where a, b, c and d are positive integers. - Edexcel - GCSE Maths - Question 10 - 2017 - Paper 1
Question 10
Show that \( (x + 1)(x + 2)(x + 3) \) can be written in the form \( ax^3 + bx^2 + cx + d \) where a, b, c and d are positive integers.
Worked Solution & Example Answer:Show that \( (x + 1)(x + 2)(x + 3) \) can be written in the form \( ax^3 + bx^2 + cx + d \) where a, b, c and d are positive integers. - Edexcel - GCSE Maths - Question 10 - 2017 - Paper 1
Step 1
Step 1: Expand the Expression
Answer
To show that ( (x + 1)(x + 2)(x + 3) ) can be expressed as ( ax^3 + bx^2 + cx + d ), we first need to expand the expression. We can do this by multiplying two of the factors first:
[ (x + 1)(x + 2) = x^2 + 2x + 1x + 2 = x^2 + 3x + 2 ]
Next, we will multiply this result by the remaining factor ( (x + 3) ):
[ (x^2 + 3x + 2)(x + 3) = x^3 + 3x^2 + 3x^2 + 9x + 2x + 6 = x^3 + 6x^2 + 11x + 6 ]
Step 2
Step 2: Identify Coefficients a, b, c, d
Answer
From the expanded polynomial ( x^3 + 6x^2 + 11x + 6 ), we can identify the coefficients:
- ( a = 1 )
- ( b = 6 )
- ( c = 11 )
- ( d = 6 )
Since all coefficients ( a, b, c, d ) are positive integers, we can conclude that the given expression can indeed be represented in the required form.