The function f(x)=ax^3-x^3+bx-24 has three factors. Two of these factors are x-2 and x+4. Determine the values of a and b , and then determine the other factor
If you are confused about how they jump from two equations to the answers, I'll simplify your studying in three words - Method of Elimination. You may remember it from solving linear problems in earlier years, anyway I'll explain it briefly as a refresher but it might be easier if you watch a tutorial on it. The Method of Elimination basically means to "eliminate" a variable. Take the two equations that we found by factoring
1. 8a+2b=28
2. -64a-4b=40
We need to eliminate one of them, the easiest one to zero out is b, we can do that by multiplying equation 1 by 2. See below:
8a+2b=28 X2 = 16a +4b=56
Now plain and simple add the new equation 1 to the original equation 2.
(16a +4b=56) + (-64a-4b=40) = -48a +0b = 96 or -48a=96
Solve for a which gives you -2 and then use the a value to solve for b.
Well, since two of the factors are x-2 and x+4, we can use synthetic division to find the remaining factor!
First, let's divide f(x) by (x-2):
2 | a -1 b -24
- 2a-2 2a+4 6a+8
--------------
a-2a-1 b-2a+4 6a-16
Now, let's divide the resulting polynomial by (x+4):
-4 | a-2 -1 b-2a+4 6a-16
-a+2 5a-18 -6a+24 0
-----------------------
a-3 4a-16 -6a+28 6a-16
Notice that the resulting polynomial is 6a-16, which means it should be equal to zero in order to have the third factor.
Setting 6a-16 = 0 and solving for a gives us:
6a = 16
a = 16/6
a = 8/3
Now that we know a, let's determine b using one of the previous synthetic divisions. Let's use the first one:
2 | 8/3 -1 b -24
- 16/3 32/3 52/3
----------------------
8/3 11/3 (b+32)/3 4/3
Since we know the remainder after dividing by x-2 (4/3) should be zero, we have:
4/3 = 0
Which is not true!
So, it seems there might be an error in the question. Please double-check the function f(x) given to determine if there was any mistyping or missing information.
To determine the values of a and b, we can use the given factors, x-2 and x+4, to find the third factor.
Step 1: Apply the factor theorem.
The factor theorem states that if a polynomial f(x) has a factor (x-a), then f(a) = 0. By applying this theorem, we can determine the values of a and b.
Step 2: Substitute x-2 and x+4 into the function.
For f(x) = ax^3 - x^3 + bx - 24, we substitute x-2 and x+4 into the function to get two equations:
When x = 2:
f(2) = a(2^3) - 2^3 + b(2) - 24
When x = -4:
f(-4) = a(-4^3) - (-4)^3 + b(-4) - 24
Step 3: Simplify and solve the equations.
Now we solve these two equations to determine the values of a and b.
For x = 2:
f(2) = a(8) - 8 + 2b - 24
0 = 8a - 8 + 2b - 24
8a + 2b = 32 (equation 1)
For x = -4:
f(-4) = a(-64) - (-4)^3 + b(-4) - 24
0 = -64a + 64 + 4b - 24
-64a + 4b = -40 (equation 2)
Now we have a system of linear equations (equations 1 and 2), which can be solved simultaneously.
Step 4: Solve the system of equations.
Multiply equation 1 by 32 and equation 2 by 2 to eliminate the coefficient of b.
256a + 64b = 1024 (equation 3)
-128a + 8b = -80 (equation 4)
Adding equations 3 and 4 together yields:
128a + 72b = 944 (equation 5)
Now we can solve equation 5 for 'a':
128a = 944 - 72b
a = (944 - 72b) / 128
a = (118 - 9b) / 16
Since a doesn't have a specific value, it's best to express it in terms of b.
Step 5: Find the value of b and substitute it into the equation for the factor.
To find the value of b, we can substitute the expression for a into either equation 1 or equation 2.
Let's substitute into equation 1:
8[(118 - 9b) / 16] + 2b = 32
(118 - 9b) + 8b = 32*16
118 - b = 512
b = 118 - 512
b = -394
So, the value of b is -394.
Step 6: Determine the third factor.
Now that we found the values of a and b, we can substitute them back into the original equation and simplify it.
f(x) = ax^3 - x^3 + bx - 24
f(x) = [(118 - 9b) / 16]x^3 - x^3 + (-394)x - 24
f(x) = [(118 - (9 * (-394))) / 16]x^3 - x^3 - 394x - 24
f(x) = 59x^3 - x^3 - 394x - 24
f(x) = 58x^3 - 394x - 24
Therefore, the third factor is 58x^3 - 394x - 24.
synthetic division yields remainders
f(x) = (x-2)*p(x) + 8a+2b-28
f(x) = (x+4)*q(x) + (-64a-4b-40)
so,
8a+2b = 28
64a+4b = -40
a = -2
b = 22
f(x) = (x-2)(x+4)(-2x+3)