Please help me ...Find a polynomial of degree 4 with -2 as a zero of multiplicity 2 and 0 and 3 as zeros of multiplicity 1.
Start by listing "x = " as many times as the multiplicity
of each zero, for every zero:
x = -2, x = -2, x = 0, x = 3
[Notice that I listed x = -2 twice because it has
multiplicity 2]
Get 0 on the right of each of those equations:
x + 2 = 0, x + 2 = 0, x = 0, x - 3 = 0
[Notice that x = 0 already had a 0 on the right side]
Now indicate the multiplication of the left sides
of all of those equations and set it = 0
(x + 2)(x + 2)(x)(x - 3) = 0
Now multiply that out.
Multiplying the first two factors:
(x² + 4x + 4)(x)(x - 3) = 0
Multiply the first two factor of this:
(x³ + 4x² + 4x)(x - 3) = 0
Multiply those:
x⁴- 3x³ + 4x³ - 12x² + 4x² - 12x = 0
Collect like terms:
x⁴+ x³ - 8x² - 12x = 0
The polynomial itself does not contain the = 0
on the end, so the answer is just the left side:
P(x) = x⁴+ x³ - 8x² - 12x
Edwin
