Question

Use the Fundamental Theorem of Algebra to determine which of the following equations have two solutions. Explain your reasoning in 3–5 sentences.

a. −9x+4x2+3=0
b. −2x2−5=0
c. 57x2=0
d. 0x2+7x−2=0
e. 0x3−x2+2x+4=0
(4 points)

Answer 1

The Fundamental Theorem of Algebra says a nonzero polynomial of degree n has exactly n complex roots counted with multiplicity. So (a) −9x+4x^2+3 = 4x^2−9x+3 and (e) −x^2+2x+4 are degree 2 and therefore have two solutions (both a and e in fact give two distinct real roots since their discriminants are positive). (b) −2x^2−5 is degree 2 so it has two solutions as well, but they are nonreal complex conjugates. (c) 57x^2=0 has root x=0 of multiplicity 2 (one distinct solution counted twice) and (d) 7x−2=0 is linear so has exactly one solution.