Which binomial is a factor of f(x) = x^4 – 3x^3 – 28x^2 + 36x + 144?
(x – 4)
(x + 3)
(x – 2)
(x – 6)
I think its the 4th one
What makes you think so ?
The 4th one is the answer
To determine which binomial is a factor of the given polynomial f(x), we can use a process called synthetic division.
Start by setting up the synthetic division as follows:
| 1 -3 -28 36 144
-------------------------------------
(6) |
Here, we're using the divisor (x - 6) as the synthetic division is set up. The coefficients of the polynomial f(x) are placed in the first row, and the divisor coefficients are placed beneath the line.
The first step is to bring down the 1 from the first row:
| 1 -3 -28 36 144
-------------------------------------
(6) | 1
Next, multiply the divisor by the value just brought down (1), and write the result below the next coefficient (in this case, -3):
| 1 -3 -28 36 144
-------------------------------------
(6) | 1
-6
Add these two values (-3 and -6), writing the sum (in this case, -9) below the next coefficient (-28):
| 1 -3 -28 36 144
-------------------------------------
(6) | 1 -9
-6
Repeat the previous step for each coefficient, continuing with -9 multiplied by 6:
| 1 -3 -28 36 144
-------------------------------------
(6) | 1 -9 -54
-6
And so on, until you reach the last coefficient:
| 1 -3 -28 36 144
-------------------------------------
(6) | 1 -9 -54 0
-6 0
The final row of the synthetic division represents the coefficients of the quotient polynomial. If the remainder is zero (as it is in this case), it means that the divisor (x - 6) is a factor of the polynomial f(x).
Looking at the final row: 1, -9, -54, 0, these coefficients correspond to the polynomial x^3 - 9x^2 - 54x + 0.
Since the last term of any polynomial divided by (x - 6) should be zero, we can simplify this further to x^3 - 9x^2 - 54x.
Therefore, (x - 6) is a factor of f(x) = x^4 - 3x^3 - 28x^2 + 36x + 144.
Thus, your initial intuition is correct, and (x - 6) is the correct binomial factor.