How do i find a and b from this cubic?.....maths Q

f[x] = 2x^3 + ax^2 + bx + 15

given that when f[x] is divided by [x-2] the remainder is 35. [x+3] is a factor of f[x].

How would I find a and b?????

a = 3
b = - 4


i found it setting up synthetic division

the coefficients would be 2 a b 15

then, we would divide 2 into it using synthetic division, which results in:

4a + 2b + 31, which is said to equal 35

setting it equal to 35 we can simplify getting:

4a + 2b + 31 = 35 subtract 31 from both sides
4a + 2b = 4 divide both sides by 2
2a + b = 2

now, using synthetic division to factor out 3, we get:

9a -3b - 39, which is said to equal zero. set it equal to zero and simplify

9a - 3b -39 = 0 add 39 to both sides
9a - 3b = 39 divide all sides by 3
3a -b = 13

we now have two equations:

2a + b = 2
3a -b = 13 add them up, the b's cancel out
----------------
5a = 15 divide both sides by 5

a = 3

plug this back into one of the equations to find b

2a + b = 2 sub in 3 for a
2(3) + b = 2
6 + b = 2 subtract 6 from both sides
b = -4

when we put these into the function, we get:

f(x) = 2x^3 +3x² - 4x + 15

if you plug in 2 and 3, you will find that 2 results in 35 and 3 results in 0. which shows we have solved it correctly

sorry that i couldn't adequately show the synthetic division. it's hard to show

basically, u take the coefficients from the function. and then off to the side, you put what you want to "divide" into it. in this case, if they say x - 2, set it equal to zero to find what you are dividing synthetically, which is 2. the x + 3, you are synthetically dividing -3. you take down the first coeffcient, in this case 2

then you multiply it by what you are dividing out, lets say the 2:

2(2) = 4 then you add that to the next coeffecient, which is a

a + 4 then multiply by the 2 you are dividng out again