What we learned in the previous section can be expanded to higher degree polynomials. Let's write an n^th degree polynomial equation in a general form:

a₀xⁿ + a₁x^n-1 + a₂x^n-2 + ... + a_n-1x + a_n = 0

The sum of the roots of this polynomial is going to be -a₁/a₀.

And the product of the roots? Well, that's a little more complicated. If n is even, then the product of the roots is a_n/a₀. If, on the other hand, n is odd, then the product of the roots is -a_n/a₀.

Example: Find the sum of the roots of x³ + 3x² - 4xx - 12 = 0

Solution: The sum of the roots is -3/1 = -3.

Example: Find the sum of the roots of x⁴ - 7x - 6 = 0.

Solution: In this case there is no x³ term, so we think of it as 0x³. Thus, the sum of the roots is -0/1 = 0.

Example: Find the product of the roots of 2x⁵ + 3x³ - 1 = 0.

Solution: Since this is an odd degree polynomial equation, the product of the roots is the opposite of the constant term divided by the leading coefficient: 1/2.

Example: Find the product of the roots of 81x⁴ - 1 = 0.

Solution: Since this is an even degree polynomial equation, the product of the roots is the constant divided by the leading coefficient: -1/81.

1.

What is the product of the roots of x³ + 2x² - 4x + 10 = 0

2.

What is the product of the roots of 3x⁴ - 8 = 0

3.

What is the sum of the roots of x⁵ - 2 = 0

4.

What is the product of the roots of 4x⁴ - 3x³ + 2x² -x?

5.

The leading coefficient of a cubic polynomial is 2. The zeroes are 3, 4, and 5. What is the constant term of the polynomial?

6.

The roots of a cubic equation are all consecutive integers. The first two terms of the equation are 2x³ - 48x². Find the constant term of the equation.

7.

The roots of a cubic equation are 1, -2, and 3. The sum of the leading coefficient and the constant term is 21. What is the constant term?

8.

The roots of a quartic equation with integer coefficients are all distinct prime numbers. What is the smallest possible value for the constant term?