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Algebra

Polynomial Equations

What are examples of polynomials with one root and no roots?

The following is an example of a polynomial with only one root:

x2 + 6x + 9 = 0

(x2 + 6x + (6/2)2) - (6/2)2 + 9 = 0

(x + 3)2 - 9 + 9 = 0

(x + 3) 2 = 0

x + 3 = 0

x = -3, or the polynomial has only one root x = -3

But not all polynomials have roots. The following is an example of a polynomial with no root:

2x2 - 6x + 8 = 0

(½)(2x2- 6x + 8) = (½)0

x2 - 3x + 4 = 0

(x2 - 3x + (-3/2)2) - (-3/2)2 + 4 = 0

(x - 3/2)2- 9/4 + 4 = 0

(x - 3/2)2 + 7/4 = 0

(x - 3/2)2 = -7/4

Because a real number squared is greater than or equal to 0, that means (x - 3/2)2 will always be greater than or equal to 0. Therefore, the answer can’t be -7/4, a negative number, and there are no real roots for this polynomial.