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Show that $(2x + 1)(x + 3)(3x + 7)$ can be written in the form $ax^3 + bx^2 + cx + d$ where $a$, $b$, $c$ and $d$ are integers - Edexcel - GCSE Maths - Question 20 - 2019 - Paper 3
Question 20
Show that $(2x + 1)(x + 3)(3x + 7)$ can be written in the form $ax^3 + bx^2 + cx + d$ where $a$, $b$, $c$ and $d$ are integers. Solve $(1 - x)^2 < \frac{9}{25}$.
Worked Solution & Example Answer:Show that $(2x + 1)(x + 3)(3x + 7)$ can be written in the form $ax^3 + bx^2 + cx + d$ where $a$, $b$, $c$ and $d$ are integers - Edexcel - GCSE Maths - Question 20 - 2019 - Paper 3
Step 1
Show that $(2x + 1)(x + 3)(3x + 7)$ can be written in the form $ax^3 + bx^2 + cx + d$
Answer
To express the product in the form , we will perform the multiplication systematically.
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Multiply the first two factors:
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Now multiply this result by the third factor:
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Use the distributive property (FOIL):
Combining like terms gives:
Simplifying further leads to: .
Thus, comparing with the form , we have:
- .
Step 2
Solve $(1 - x)^2 < \frac{9}{25}$
Answer
To solve the inequality , we start by taking the square root of both sides:
This leads us to two inequalities:
For the first inequality:
Subtracting 1 from both sides:
Multiplying by -1 (reversing the inequality):
For the second inequality:
Subtracting 1 gives:
Multiplying by -1 (reversing the inequality):
Combining both parts, we find: .