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 product
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Answer
To show that (x+1)(x+2)(x+3) can be expressed in the desired form, we first expand the product. We can group the factors as follows:
First multiply the first two factors:
(x+1)(x+2)=x2+2x+x+2=x2+3x+2
Next, multiply this result by the third factor:
(x2+3x+2)(x+3)
Step 2
Step 2: Complete the multiplication
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Answer
We will distribute each term in (x2+3x+2) by (x+3):
Multiply x2 by (x+3):
x2imes(x+3)=x3+3x2
Multiply 3x by (x+3):
3ximes(x+3)=3x2+9x
Multiply 2 by (x+3):
2imes(x+3)=2x+6
Combining all these terms gives:
x3+3x2+3x2+9x+2x+6
Step 3
Step 3: Simplify the expression
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Answer
Now we combine like terms:
x3+(3x2+3x2)+(9x+2x)+6
This simplifies to:
x3+6x2+11x+6
This is now in the form ax3+bx2+cx+d, where:
a=1
b=6
c=11
d=6
Since all coefficients are positive integers, we have successfully shown that the expression can be represented in the required form.
