What are the additional linear factors of x^3−6x^2+11x−6 if x−3 is a factor? (1 point)
a. (x-2)(x+1)
b. (x+2)(x-1)
c. (x-2)(x-1)
d. (x+2)(x+1)
To find the additional linear factors of a polynomial when one factor is given, you can use long division or synthetic division. Since the factor x - 3 is given, we can use synthetic division to find the quotient.
Using synthetic division, we set up the problem as follows:
3 | 1 -6 11 -6
___________________
| 3 -9 6
|___________________
1 -3 2 0
The remainder is 0, and the coefficients of the quotient are 1, -3, and 2.
Therefore, the quotient is x^2 - 3x + 2.
The additional linear factors of the polynomial x^3 - 6x^2 + 11x - 6 are the solutions to the quadratic equation x^2 - 3x + 2 = 0.
To factor the quadratic equation, we find the roots by factoring:
(x - 2)(x - 1) = 0
So, the additional linear factors are:
(x - 2)(x - 1)
Therefore, the correct answer is c. (x-2)(x-1).
To find the additional linear factors, we need to perform polynomial long division with x-3 as the divisor.
Dividing x^3-6x^2+11x-6 by x-3:
x^2 - 3x + 2
___________________
x - 3 | x^3 - 6x^2 + 11x - 6
- (x^3 - 3x^2)
________________
-3x^2 + 11x
+ (3x^2 - 9x)
________________
2x - 6
- (2x - 6)
________________
0
The quotient is x^2 - 3x + 2.
Therefore, the additional linear factors are (x-2)(x-1).
Therefore, the correct answer is c. (x-2)(x-1).
To find the additional linear factors of a polynomial, you can use long division or synthetic division to divide the polynomial by the given factor.
In this case, the given factor is x - 3. To perform long division, set up the division problem as follows:
x^2 + 3x + 2
x - 3 │ x^3 - 6x^2 + 11x - 6
To begin the division, ask yourself: "What do I need to multiply x by to obtain x^3?" The answer is x^2. So write x^2 above the line.
x^2 + 3x + 2
x - 3 │ x^3 - 6x^2 + 11x - 6
- (x^3 - 3x^2)
Now subtract (x^2)(x - 3) from the top line:
x^2 + 3x + 2
x - 3 │ x^3 - 6x^2 + 11x - 6
- (x^3 - 3x^2)
--------------
-3x^2 + 11x
Next, ask yourself: "What do I need to multiply x by to obtain -3x^2?" The answer is -3x. So write -3x above the line.
x^2 + 3x + 2
x - 3 │ x^3 - 6x^2 + 11x - 6
- (x^3 - 3x^2)
--------------
-3x^2 + 11x
- (-3x^2 + 9x)
Now subtract (-3x)(x - 3) from the top line:
x^2 + 3x + 2
x - 3 │ x^3 - 6x^2 + 11x - 6
- (x^3 - 3x^2)
--------------
-3x^2 + 11x
- (-3x^2 + 9x)
---------------
2x - 6
At this point, the remainder is 2x - 6. Since the degree of the remainder is less than the degree of the divisor (x - 3), long division stops here.
Therefore, the additional linear factors are (x^2 + 3x + 2), which can be factored as (x + 2)(x + 1).
So, the correct answer is:
d. (x + 2)(x + 1)