polynomial equation with intercepts
(-3/2,0)
(1,0)
(5/2,0)
(0,4/3)
To find a polynomial equation with the given intercepts, we can use the fact that the intercepts occur when the function equals zero.
Let's assume the polynomial equation has the form:
f(x) = a(x - r)(x - s)(x - t)
where r, s, and t are the intercepts, and a is a constant.
Given the intercepts (-3/2, 0), (1, 0), and (5/2, 0), we have the equations:
a(-3/2 - r)(-3/2 - s)(-3/2 - t) = 0 (equation 1)
a(1 - r)(1 - s)(1 - t) = 0 (equation 2)
a(5/2 - r)(5/2 - s)(5/2 - t) = 0 (equation 3)
Additionally, we know that the point (0, 4/3) lies on the curve, so we have:
f(0) = 4/3
Substituting x = 0 into the equation f(x) = a(x - r)(x - s)(x - t), we get:
f(0) = a(0 - r)(0 - s)(0 - t) = 4/3
Since we have four unknowns (a, r, s, and t), we can solve these four equations simultaneously to find the values.
Using equation 1, we can write it as:
(-3/2 - r)(-3/2 - s)(-3/2 - t) = 0
Expanding this equation gives:
(-3/2)(-3/2)(-3/2) - (-3/2)(-3/2)(r + s + t) + (-3/2)(r)(s)(t) = 0
Simplifying further:
27/8 + 9/4(r + s + t) - (3/2)(rst) = 0
Similarly, expanding equation 2 and equation 3, we get:
27/8 + 9/4(r + s + t) - (3/2)(rst) = 0 (equation 4)
15/8 + 15/4(r + s + t) - (15/2)(rst) = 0 (equation 5)
Now we have two equations (equation 4 and equation 5) with two unknowns (r + s + t and rst). We can solve these two equations to find the values of r, s, and t.
After solving for r, s, and t, we can substitute these values into equation 1, equation 2, and equation 3 to find the constant 'a'.
Finally, we can write the polynomial equation in the form:
f(x) = a(x - r)(x - s)(x - t)