In stuck at (x^3+1)(x+1)
Can someone explain how I countinue, please be specific
i need help factoring x^4+x^3+×+1?
2015-09-01 7:59 pm
回答 (4)
2015-09-01 8:34 pm
The expression
= x⁴ + x³ + x + 1
= (x⁴ + x³) + (x + 1)
= x³(x + 1) + (x + 1)
= (x + 1)(x³ + 1)
It is known that: a³ + b³ = (a + b)(a² - ab + b²)
Hence, x³ + 1
= x³ + 1³
= (x + 1)(x² - x + 1)
The expression
= (x + 1) [(x + 1)(x² - x + 1)]
= (x + 1)²(x² - x + 1)
= x⁴ + x³ + x + 1
= (x⁴ + x³) + (x + 1)
= x³(x + 1) + (x + 1)
= (x + 1)(x³ + 1)
It is known that: a³ + b³ = (a + b)(a² - ab + b²)
Hence, x³ + 1
= x³ + 1³
= (x + 1)(x² - x + 1)
The expression
= (x + 1) [(x + 1)(x² - x + 1)]
= (x + 1)²(x² - x + 1)
2015-09-01 8:23 pm
x^3 [ x + 1 ] + [ x + 1 ]
[ x + 1 ] [ x^3 + 1 ]
[ x + 1 ] [ x^3 + 1 ]
2015-09-01 8:10 pm
= x^3(x + 1) + x + 1
= (x^3 + 1)(x + 1)
= (x + 1)[x^2(x + 1) - x(x + 1) + x + 1]
= (x + 1)^2(x^2 - x + 1)
= (x^3 + 1)(x + 1)
= (x + 1)[x^2(x + 1) - x(x + 1) + x + 1]
= (x + 1)^2(x^2 - x + 1)
2015-09-01 8:09 pm
The factors of the sum of two cubes are (a+b)(a^2-ab+b).
In your problem, the factors of (x^3 + 1) are (x+1) and (x^2-x+1).
So the factors of your original expression are
(x+1)(x+1)(x^2-x+1).
In your problem, the factors of (x^3 + 1) are (x+1) and (x^2-x+1).
So the factors of your original expression are
(x+1)(x+1)(x^2-x+1).
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