Decompose the polynomial x^3−2x^2−5x+6
into its linear factors, given that x−1
is a factor.
Let's use synthetic division to find the other factor of the polynomial.
1 │ 1 -2 -5 6
──────────────
1 -1 -6
───────────
1 -3 0
The remainder is 0, which means that (x - 1) is a factor of the polynomial. The quotient is x^2 - 3.
Therefore, the polynomial x^3 - 2x^2 - 5x + 6 can be decomposed as (x - 1)(x^2 - 3).
To decompose the polynomial into its linear factors, given that x-1 is a factor, we can use polynomial long division or synthetic division to divide the polynomial by (x-1).
Let's use polynomial long division:
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x - 1 | x^3 - 2x^2 - 5x + 6
To start, divide the first term, x^3, by (x-1), which gives x^2.
x^2
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x - 1 | x^3 - 2x^2 - 5x + 6
Now, multiply (x-1) by x^2, resulting in x^3 - x^2. Subtract this from the original polynomial:
x^2
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x - 1 | x^3 - 2x^2 - 5x + 6
- (x^3 - x^2)
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- x^2 - 5x + 6
Bring down the next term, -5x:
x^2 - x
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x - 1 | x^3 - 2x^2 - 5x + 6
- (x^3 - x^2)
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- x^2 - 5x + 6
- (-5x)
Next, divide - x^2 by (x-1), which gives -x:
x^2 - x
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x - 1 | x^3 - 2x^2 - 5x + 6
- (x^3 - x^2)
________________________
- x^2 - 5x + 6
- (-5x)
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- 4x + 6
Finally, divide -4x by (x-1), resulting in -4:
x^2 - x - 4
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x - 1 | x^3 - 2x^2 - 5x + 6
- (x^3 - x^2)
________________________
- x^2 - 5x + 6
- (-5x)
________________________
- 4x + 6
- (-4x)
________________________
+10
Therefore, (x^3 - 2x^2 - 5x + 6)/(x-1) = (x^2 - x - 4) remainder 10.
Hence, we have factored the polynomial x^3 - 2x^2 - 5x + 6 using x-1 as a factor:
x^3 - 2x^2 - 5x + 6 = (x-1)(x^2 - x - 4) + 10.
To decompose the polynomial x^3 - 2x^2 - 5x + 6 into linear factors, we need to use synthetic division or long division to perform polynomial division.
Given that x - 1 is a factor, we can use synthetic division to divide x^3 - 2x^2 - 5x + 6 by x - 1:
1 | 1 -2 -5 6
- 1 -1 -4
______________
1 -1 -6 2
The result of synthetic division gives us the coefficients of the quotient polynomial: 1x^2 - 1x - 6, and a remainder of 2.
Therefore, the original polynomial x^3 - 2x^2 - 5x + 6 can be factored as (x - 1)(x^2 - x - 6).
To factor the quadratic polynomial x^2 - x - 6 in the second factor further, we need to find two linear factors whose product equals -6 and whose sum equals -1, which are -3 and 2. Therefore, we can factor x^2 - x - 6 as (x - 3)(x + 2).
Therefore, the complete factorization of the polynomial x^3 - 2x^2 - 5x + 6 is (x - 1)(x - 3)(x + 2).