A cubic polynomial function fx has leading coefficient2 and

A cubic polynomial function f(x) has leading coefficient-2 and intercepts the y-axis at -7. If f(l) = f(2) = -1, find f(-2) and write the complete function. f(x) = f(-2) =

Solution

Let, complete f(x) = ax³+bx²+cx+d

Now, given that leading coefficient is -2. So the value of a is -2.

It intercepts y-axis at -7 and we know that on the y-axis, x is always 0.

So now, a(0)³+b(0)²+c(0)+d = -7

d = -7

Now, f(1) = -1

a(1)³+b(1)²+c(1)+d = -1

Put a = -2, d = -7

(-2)(1)+b(1)+c(1)+(-7) = -1

-2+b+c-7 = -1

b+c-9 = -1

b+c = 8 _________(1)

Now,. f(2) = -1

a(2)³+b(2)²+c(2)+d = -1

Put a = -2, d = -7

(-2)(2)³+b(2)²+c(2)+(-7) = -1

-16+4b+2c-7 = -1

4b+2c = 22 _________(2)

Now from (1) and (2)------

------ eq.(2) - 2× eq.(1)

(4b+2c)-2×(b+c) = 22 - 2×8

2b = 22-16 = 6

b = 3

From eq. (1) -------

(3)+c = 8

c = 8-3 = 5

So complete f(x) = -2x³+3x²+5x-7

Now f(-2) = -2(-2)³+3(-2)²+5(-2)-7

f(-2) = 16+12-10-7

f(-2) = 28-17

f(-2) = 11