Answer:
Explanation:
The zeros of a polynomial function are the values of x for which the function equals zero. For the given polynomial function p(x) = (2x - 1)(5x + 7)(8x + 9), the zeros can be found by setting each factor equal to zero and solving for x:
2x - 1 = 0, x = 1/2
5x + 7 = 0, x = -7/5
8x + 9 = 0, x = -9/8
Therefore, the zeros of the function are x = 1/2, x = -7/5, and x = -9/8.
In standard form, the polynomial function p(x) can be written as:
p(x) = 80x^3 + 122x^2 - 127x - 63
The relationship between the zeros of p(x) and its coefficients can be seen in Vieta's formulas. Vieta's formulas state that for a polynomial function of degree n with roots r1, r2, ..., rn, the coefficients of the polynomial can be expressed as:
a0 = (-1)^n * p0
a1 = (-1)^(n-1) * p1 / p0
a2 = (-1)^(n-2) * p2 / p0
...
an-1 = (-1) * pn-1 / p0
an = pn / p0
where p0 is the coefficient of the highest degree term (the leading coefficient), and the pi are the elementary symmetric polynomials, which are given by:
p1 = r1 + r2 + ... + rn
p2 = r1r2 + r1r3 + ... + rn-1rn
...
pn-1 = r1r2...rn-1 + r1r2...rn-2 + ... + rn-2rn-1
pn = r1r2...rn
Using Vieta's formulas, we can see that for the polynomial function p(x) given above, the coefficients are related to the zeros as follows:
a0 = -63
a1 = -127
a2 = 122
a3 = 80
And we can also see that:
a0 = (-1)^3 * p0 = -63
a1 = (-1)^(3-1) * p1 / p0 = -127
a2 = (-1)^(3-2) * p2 / p0 = 122
a3 = (-1)^(3-3) * p3 / p0 = 80
Therefore, the coefficients of the polynomial are related to the zeros through Vieta's formulas, which express the coefficients as functions of the zeros, and vice versa.