# Higher Degree Coefficients and Roots

Reference > Mathematics > Algebra > PolynomialsWhat 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:

a_{0}x^{n} + a_{1}x^{n-1} + a_{2}x^{n-2} + ... + a_{n-1}x + a_{n} = 0

The sum of the roots of this polynomial is going to be -a_{1}/a_{0}.

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}/a_{0}. If, on the other hand, n is odd, then the product of the roots is -a_{n}/a_{0}.

**Example:** Find the sum of the roots of x^{3} + 3x^{2} - 4xx - 12 = 0

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

**Example:** Find the sum of the roots of x^{4} - 7x - 6 = 0.

**Solution:** In this case there is no x^{3} term, so we think of it as 0x^{3}. Thus, the sum of the roots is -0/1 = 0.

**Example:** Find the product of the roots of 2x^{5} + 3x^{3} - 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 81x^{4} - 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.

## Questions

^{3}+ 2x

^{2}- 4x + 10 = 0

^{4}- 8 = 0

^{5}- 2 = 0

^{4}- 3x

^{3}+ 2x

^{2}-x?

^{3}- 48x

^{2}. Find the constant term of the equation.