(a) Solve x^2+8X-9=0 by factoring. Find the sum and the product of the zeros.
(b) Suppose x^2+bx+Chas zeros r1 and r2. Write the polynomial in factored form. Multiply the factored form out.
What relationships do you find between this product and the coefficients in the original polynomial?
(c) Factor f(x)=x^3-x^2-10x-8 using the techniques of this module. Find the sum of the zeros and the product of the
zeros. Now find the sum of all the double products of the zeros (i.e., find r1r2+r1r3_r2r3 . What is the relationship
between these three values and the coefficients of the polynomial?
(d) Suppose the monic (leading coefficient 1) cubic polynomial function f(x)=x^3+bx^2+cx+d has zeros r1, r2, and
r3. Write the polynomial in factored form and multiply the factors. Write the relationships between the coefficients of
each form of the polynomial.
(e) Suppose the monic quartic polynomial function f(x)=x^4+bx^3+cx^2+dx+e has zeros r1, r2, r3, and r4. Repeat
part (d) for this polynomial function.
(f) What relationship do you find between the zeros of the polynomial function and the coefficients of the monic
polynomial function? Can you state some general formulas? Will these formulas always work? Explain. Why do you
think having formulas like this would be useful? Explain.
(a) x^2 + 8x - 9 = 0
x^2 - x + 9x - 9 = 0
x(x - 1) + 9(x - 1) = 0
(x + 9)(x - 1) = 0
x = {-9, 1}
Sum of the zeros = -9 + 1 = -8 and Product of the zeros = -9 * 1 = -9
(b) x^2 + bx + c can be written in the factored form as (x - r1)(x - r2)
x^2 - r1x - r2x + r1r2
x^2 - (r1 + r2)x + r1r2
 r1 + r2 = -b and r1r2 = c
(c) Let p(x) = x^3 - x^2 - 10x - 8
By inspection, we find that p(4) = 0  (x - 4) is a factor of p(x)
On dividing p(x) by (x - 4) we get a quadratic factor x^2 + 3x + 2, which is (x + 1)(x + 2)
 x^3 - x^2 - 10x - 8 = (x + 2)(x + 1)(x - 4)
p(x) = 0 gives the roots as {-2, -1, 4}
Sum of the zeros = -2 - 1 + 4 = 1
Product of the zeros = -2 * -1 * 4 = 8
Sum of the products taken in pairs = -2 * -1 + -1 * 4 + -2 * 4 = -10
We find that sum of the zeros = -coefficient of x^2, Product of the zeros = -constant term and Sum of the products of
zeros taken in pairs = coefficient of x
(d) x^3 + bx^2 + cx + d can be written in the factored form as (x - r1)(x - r2)(x - r3)
x^3 - (r1 + r2 + r3)x^2 + (r1r2 + r2r3 + r3r1)x - r1r2r3
 r1 + r2 + r3 = -b, r1r2 + r2r3 + r3r1 = c and r1r2r3 = d
(e) x^4 + bx^3+ cx^2 + dx + e can be written in the factored form as (x - r1)(x - r2)(x - r3)(x - r4)
x^4 - (r1 + r2 + r3 + r4)x^3 + (r1r2 + r1r3 + r1r4 + r2r3 + r2r4 + r3r4)x^2 - (r1r2r3 + r1r2r4 + r1r3r4 + r2r3r4)x + r1r2r3r4
 r1 + r2 + r3 + r4 = -b, r1r2 + r1r3 + r1r4 + r2r3 + r2r4 + r3r4 = c, r1r2r3 + r1r2r4 + r1r3r4 + r2r3r4 = -d and r1r2r3r4 = e
(f) From the above exercises, we can generalize and say that there exist relationships between the sum and products of
the roots and the coefficients of a polynomial function. We can make the following observations:
(i) A polynomial has degree n, there will be n zeros, n terms in the sum of the zeros, nC2 terms in the sum of the
products taken in pairs, nC3 terms in the sum of the products taken in 3’s, … and n terms in the product of the zeros.
(ii) Sum of the zeros = -coefficient of x^(n - 1), sum of the products of the zeros taken in pairs = coefficient of x^(n - 2),
sum of the products of the zeros taken in 3’s = -coefficient of x^(n - 3) …, (Note the - and + signs alternating)