Decompose the polynomial x^3−2x^2−5x+6 into its linear factors, given that x−1 is a factor.
Since x - 1 is a factor of the polynomial, we know that (x - 1) divides evenly into the polynomial when we perform polynomial long division.
Performing polynomial long division:
___________________
(x - 1) | x^3 - 2x^2 - 5x + 6
- (x^3 - x^2)
------------
- x^2 - 5x
+ (x^2 - x)
------------
- 4x + 6
+ (4x - 4)
------------
2
After performing polynomial long division, we have:
x^3 - 2x^2 - 5x + 6 = (x - 1)(x^2 - x - 4) + 2
So, the polynomial x^3 - 2x^2 - 5x + 6 can be factored as:
(x - 1)(x^2 - x - 4) + 2
To find the other linear factors, we can perform polynomial long division.
We are given that x−1 is a factor of the polynomial x^3−2x^2−5x+6.
We start the polynomial long division by dividing x^3−2x^2−5x+6 by x−1.
________________
x − 1 | x^3 − 2x^2 − 5x + 6
To begin, we divide x^3 by x, which gives us x^2.
________________
x − 1 | x^3 − 2x^2 − 5x + 6
- (x^3 - x^2)
Next, we multiply (x−1) by (−x^2) to get −x^3 + x^2.
________________
x − 1 | x^3 − 2x^2 − 5x + 6
- (x^3 - x^2)
________________
- x^2
We subtract −x^2 from −2x^2 to get −x^2.
________________
x − 1 | x^3 − 2x^2 − 5x + 6
- (x^3 - x^2)
________________
- x^2
+ x^2
Now, we bring down the −5x term.
________________
x − 1 | x^3 − 2x^2 − 5x + 6
- (x^3 - x^2)
________________
- x^2
+ x^2
_______________
- 5x
We divide −5x by x, which gives us −5.
________________
x − 1 | x^3 − 2x^2 − 5x + 6
- (x^3 - x^2)
________________
- x^2
+ x^2
_______________
- 5x
+ 5x
_______________
0
The remainder is 0, which means x−1 is a factor of x^3−2x^2−5x+6.
To find the other factors, we can rewrite the polynomial:
x^3−2x^2−5x+6 = (x−1)(x^2+x−6)
Therefore, the polynomial x^3−2x^2−5x+6 can be decomposed into the linear factors as:
x^3−2x^2−5x+6 = (x−1)(x^2+x−6)