Answer:
x-intercepts = 1, 2, and 3 and y-intercept = -6.
Step-by-step explanation:
f(x) = x^3 − 6x^2 + 11x − 6 is a cubic polynomial, which should have a maximum of 3 roots/factors. One of the factor is given, which is (x-3). In order to find the other factors, the method of synthetic division will be used:
Column 1 2 3 4 5
3 | 1 -6 11 -6 (Row 1)
______|________3__-9___6 (Row 2)
| 1 -3 2 0 (Row 3)
The method requires to write the coefficients of all the powers in the Row 1 on the right hand side of the table. Put the factor x=3 on the left hand side of the table (Row 1 Column 1). The first step involves the second column elements to be added. Since there is only 1, thus there will be 1 below the line. Next step involves multiplying 1 with the factor 3 (which is 3) and this will be the element for row 2 column 3. The add the elements of the column 3. The answer comes out to be -3. Repeat the above steps for the columns 4 and 5. The sum of the column 5 (the last column) should be 0, which is the case. The elements of the last row are actually the coefficients of the quadratic equation which is resultant from dividing f(x) by (x-3). Thus:
x^2 - 3x + 2 = 0.
Using mid-term breaking:
x^2 - 2x - x + 2 = 0.
x(x-2) - 1(x-2) = 0.
(x-1)*(x-2) = 0
x-1 = 0 or x-2 = 0. This implies that x = 1 and 2 (the remaining x-intercepts).
For y-intercept, simply calculate f(0).
f(0) = 0^3 − 6(0^2) + 11(0) − 6 = -6.
Therefore, x-intercepts = 1, 2, and 3 and y-intercept = -6
