Sign Changes (C)

From calculus we have the following useful result.

Theorem 2728

If p(z) is a polynomial with real coefficients, a < b and the sign of p(a) is different from the sign of p(b) then p has a zero between a and b.

We also have some bits of common sense:

• If all of the exponents in the polynomial p(z) are even or all of the exponents are odd and p(a) = 0 then p(-a) = 0.
• If all of the coefficents of the polynomial have the same sign, then the polynomial has no positive roots.
• If all coefficient of the even powered terms of p(z) are positive and all the odd powered terms have coefficients which are negative, or vice-versa, then the polynomial has no negative roots.

Exercises

1.

Find all the zeros of each of the following polynomials:

(a)

x³-6x²+11x-6;

(b)

x⁴-7x³+17x²-17x + 6;

(c)

10x³-37x²+37x-6;

(d)

5x⁴ -7x³ - 3x² + 7x-2;

(e)

54x³-21x²-7x+2;

(f)

x⁴+4x³ + 4x² - 4x - 5;

2.

Suppose that p(z) = z⁴ + 3z³ + 2z² + 3z + 1. Show that p(i) = 0 and then find all the zeros of p.