Can anyone help with this cal 1 problem? I want an explanation, not just the answer, please.?

Question

Find an expression for a cubic function f if f(1) = 15 and f(-2) = f(0) = f(2) = 0.

f(x)=?

And the answer needs to be in the form... ax^3+bx^2+cx+d (ex 4x^3+5x^2+x+1)

Answer 1

f(x) = ax³ + bx² + cx + d

Substitute for x and f(x) the four values given

f(1) = a + b + c + d = 15

f(0) = d = 0

f(2) = 8a + 4b + 2c + d = 0

f(-2) = -8a +4b -2c + d = 0

Then you have 4 linear equations in 4 unknowns (a, b, c, d)

which you can solve.

d = 0, so eliminate d from others

8a + 4b + 2c = 0 [i]

-8a + 4b - 2c = 0 [ii]

8b = 0  [ add i and ii ]

b = 0

16a + 4c = 0  [ subtract i and ii ]

a + c = 15

a = 15 - c

16 (15 - c) + 4c = 0

240 - 12c = 0

c = 20

a = -5

f(x) = -5 x³ + 20 x

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