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Higher Degree Coefficients and Roots

Reference > Mathematics > Algebra > Polynomials
 

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

a0xn + a1xn-1 + a2xn-2 + ... + an-1x + an = 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 an/a0. If, on the other hand, n is odd, then the product of the roots is -an/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.

Questions

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?
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