The function g(x)=ax^3−x^2+x−24 has three factors. Two of these factors are 𝑥−2 and 𝑥+4. Determine the values of a and b then determine the other factor.
wondering the location of b.
Sorry! its g(x)=ax^3−x^2+bx−24
ax^3−x^2+bx−24
= (x-2)(x+4)(ax + c)
the ax^3 can only come from the multiplication of the first terms of each of the binomials,
(x)(x)(ax) = 3x^3 ----> a = 3
the -24 can only come from the multiplication of the last terms of each of the binomials,
(-2)(4)(c) = -24 ----> c = 3
thus: ax^3−x^2+bx−24
= (x-2)(x+4)(3x+3)
expand this, then match up the terms ending in x to find b
forget the first part of my solution, I read it as
3x^3−x^2+bx−24
we know : f(x) = ax^3−x^2+bx−24
since x-2 is a factor, f(2) = 0
f(2) = 8a - 4 + 2b - 24 = 0
8a + 2b = 28
4a + b = 14 **
f(-4) = 0
-64a - 16 - 4b - 24 = 0
-64a - 4b = 40
16a+ b = -10 ***
subtract ** from ***
12a = -24
a = -2
in **
-8 + b = 14
b = 22
but when i expand this i get 3x^3+9x^2-18x-24 and this doesn't work with the equation given as x^2 didn't have an coefficient and b was supposed to be positive.
lol! don't read my comment! i didn't see your new one before i posted mine!
how is ** - *** give - 24 ? shouldn't it be 4?
nvm got it! :) u did substitution right?
To find the other factor, we need to divide the given function by the two known factors: (𝑥−2) and (𝑥+4).
The first step is to set up the division:
________________________
(𝑥−2)(𝑥+4) | 𝑎𝑥^3 - 𝑥^2 + 𝑥 - 24
To perform the division, we can follow these steps:
1. Divide the first term of the dividend (𝑎𝑥^3) by the first term of the divisor (𝑥), which gives us 𝑎𝑥^2.
2. Multiply the divisor by 𝑎𝑥^2, which gives us (𝑎𝑥^2)(𝑥−2) = 𝑎𝑥^3−2𝑎𝑥^2.
3. Subtract this product from the original dividend to get the new dividend:
(𝑎𝑥^3 - 𝑥^2 + 𝑥 - 24) - (𝑎𝑥^3−2𝑎𝑥^2) = (-𝑥^2 + 𝑥 - 24 + 2𝑎𝑥^2)
= (2𝑎𝑥^2 - 𝑥^2 + 𝑥 - 24)
4. Bring down the next term from the original dividend, which is 𝑥:
(𝑥−2)(𝑥+4) | (2𝑎𝑥^2 - 𝑥^2 + 𝑥 - 24)
↑
𝑥
5. Divide the first term of the new dividend (2𝑎𝑥^2) by the first term of the divisor (𝑥), which gives us 2𝑎𝑥.
6. Multiply the divisor by 2𝑎𝑥, which gives us (2𝑎𝑥)(𝑥−2) = 2𝑎𝑥^2-4𝑎𝑥.
7. Subtract this product from the new dividend to get the updated dividend:
(2𝑎𝑥^2 - 𝑥^2 + 𝑥 - 24) - (2𝑎𝑥^2-4𝑎𝑥) = (-𝑥^2 + 𝑥 - 24 + 4𝑎𝑥)
= (4𝑎𝑥 - 𝑥^2 + 𝑥 - 24)
8. Bring down the last term from the original dividend, which is -24:
(𝑥−2)(𝑥+4) | (4𝑎𝑥 - 𝑥^2 + 𝑥 - 24)
↑
-24
9. Divide the first term of the updated dividend (4𝑎𝑥) by the first term of the divisor (𝑥), which gives us 4𝑎.
10. Multiply the divisor by 4𝑎, which gives us (4𝑎)(𝑥−2) = 4𝑎𝑥-8𝑎.
11. Subtract this product from the updated dividend to get the final remainder:
(4𝑎𝑥 - 𝑥^2 + 𝑥 - 24) - (4𝑎𝑥-8𝑎) = (-𝑥^2 + 𝑥 - 24 + 8𝑎)
= (-𝑥^2 + 𝑥+8𝑎 - 24)
Now, to find the last factor, we set the final remainder equal to zero and solve for 𝑥:
-𝑥^2 + 𝑥+8𝑎 - 24 = 0
This is a quadratic equation, and we can use the quadratic formula to find its roots.
The quadratic formula is given by:
𝑥 = (-𝑏 ± √(𝑏^2 - 4𝑎𝑐))/(2𝑎)
Comparing the coefficients of the quadratic equation, we have:
𝑎 = -1,
𝑏 = 1, and
𝑐 = 8𝑎 - 24.
Plugging these values into the quadratic formula, we get:
𝑥 = (-(1) ± √((1)^2 - 4(-1)(8𝑎 - 24)))/(2(-1))
Simplifying further:
𝑥 = (-1 ± √(1 + 32𝑎 - 96))/(2)
Now, we need to solve this quadratic equation for 𝑥.