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Can anyone explain how to factorise this? Thanks.
Date Posted: 3 years ago
Views: 314
See 1 Answer
You will first have to factorise out x² from the three terms in front.
x⁴ - 4x³ + 4x² - (x - 2)²
= x²(x² - 4x + 4) - (x - 2)²
Then, realise that (x² - 4x + 4) can be rewritten as (x² - 2(2)x + 2²) , which is in the form a² - 2ab + b².
Now, use the property a² - 2ab + b² = (a - b)² to rewrite the above as (x - 2)²
Continued from above :
= x²(x - 2)² - (x - 2)²
Now you can factor out (x - 2)²
= (x² - 1)(x - 2)²
Lastly, x² - 1 can be rewritten as (x - 1)(x + 1) using the property a² - b² = (a + b)(a - b)
= (x - 1)(x + 1)(x - 2)²
x⁴ - 4x³ + 4x² - (x - 2)²
= x²(x² - 4x + 4) - (x - 2)²
Then, realise that (x² - 4x + 4) can be rewritten as (x² - 2(2)x + 2²) , which is in the form a² - 2ab + b².
Now, use the property a² - 2ab + b² = (a - b)² to rewrite the above as (x - 2)²
Continued from above :
= x²(x - 2)² - (x - 2)²
Now you can factor out (x - 2)²
= (x² - 1)(x - 2)²
Lastly, x² - 1 can be rewritten as (x - 1)(x + 1) using the property a² - b² = (a + b)(a - b)
= (x - 1)(x + 1)(x - 2)²
Thanks for your explanation!
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Alternative way :
Realise that x⁴ - 4x³ + 4x² can immediately be rewritten as (x²)² - 2(x²)(2x) + (2x)²
This is also in the form a² - 2ab + b² .
So x⁴ - 4x³ + 4x² - (x - 2)²
= (x²)² - 2(x²)(2x) + (2x)² - (x - 2)²
= (x² - 2x)² - (x - 2)²
From here, factor out the x.
= (x(x - 2))² - (x - 2)²
Then use the property (ab)ⁿ = aⁿbⁿ
= x²(x - 2)² - (x - 2)²
The rest of the steps are identical to the first method's.
However, this approach might he harder to spot from the start.
Realise that x⁴ - 4x³ + 4x² can immediately be rewritten as (x²)² - 2(x²)(2x) + (2x)²
This is also in the form a² - 2ab + b² .
So x⁴ - 4x³ + 4x² - (x - 2)²
= (x²)² - 2(x²)(2x) + (2x)² - (x - 2)²
= (x² - 2x)² - (x - 2)²
From here, factor out the x.
= (x(x - 2))² - (x - 2)²
Then use the property (ab)ⁿ = aⁿbⁿ
= x²(x - 2)² - (x - 2)²
The rest of the steps are identical to the first method's.
However, this approach might he harder to spot from the start.
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