find the value of k for which x - 3 is a factor of f(x) =4x³ +6x² - 9x + 2k, hence find the remainder when f(x) is divided by x+2
again, use the Remainder Theorem.
f(-2) = 0
To find the value of k for which x - 3 is a factor of f(x), we need to use the factor theorem. According to the factor theorem, a polynomial is divisible by x - a if and only if the polynomial evaluates to zero when the value of x is a.
So, let's substitute x = 3 into f(x) and set it equal to zero:
f(3) = 4(3)³ + 6(3)² - 9(3) + 2k
Now, simplify the expression:
f(3) = 4(27) + 6(9) - 27 + 2k
f(3) = 108 + 54 - 27 + 2k
f(3) = 135 + 2k
Since x - 3 is a factor of f(x), f(3) should be equal to zero:
135 + 2k = 0
To find the value of k, we can solve this equation:
2k = -135
k = -135/2
k = -67.5
So, the value of k for which x - 3 is a factor of f(x) is -67.5.
To find the remainder when f(x) is divided by x + 2, we can use synthetic division. The coefficients of f(x) are 4, 6, -9, and 2k. Since k = -67.5, we can substitute this value into f(x):
f(x) = 4x³ + 6x² - 9x + 2(-67.5)
f(x) = 4x³ + 6x² - 9x - 135
Using synthetic division, we divide f(x) by x + 2:
-2 | 4 6 -9 -135
| -8 4 10
---------------
4 -2 -5 -125
The remainder is -125.
Therefore, the remainder when f(x) is divided by x + 2 is -125.
To find the value of k for which x - 3 is a factor of f(x), we need to use the remainder theorem. According to the remainder theorem, if (x - 3) is a factor of f(x), then f(3) should be equal to zero.
Let's substitute x = 3 into f(x):
f(3) = 4(3)³ + 6(3)² - 9(3) + 2k
= 108 + 54 - 27 + 2k
= 135 + 2k
Since x - 3 is a factor of f(x), f(3) must be equal to zero. Therefore, we have the equation:
135 + 2k = 0
To solve for k, we subtract 135 from both sides:
2k = -135
Finally, divide both sides by 2 to isolate k:
k = -135/2
Therefore, when k = -67.5, x - 3 is a factor of f(x).
To find the remainder when f(x) is divided by x + 2, we can use the synthetic division method.
The coefficients of f(x) are 4, 6, -9, and 2k. Substituting k = -67.5 into these coefficients, we have:
f(x) = 4x³ + 6x² - 9x + 2(-67.5)
= 4x³ + 6x² - 9x - 135
Now, let's perform synthetic division using x = -2:
-2 | 4 6 -9 -135
-8 4 10
_______________________
4 -2 -5 -125
The remainder is the last number in the bottom row, which is -125.
Therefore, the remainder when f(x) is divided by x + 2 is -125.