document.write( "Question 30217: Explain why x to the 4th+2xsquared+4 has no real root while every polynomial function of degree 3 has at least 1 real root \n" ); document.write( "
Algebra.Com's Answer #16904 by Fermat(136)\"\" \"About 
You can put this solution on YOUR website!
x^4 + 2x^2 + 4
\n" ); document.write( "let u = x^2
\n" ); document.write( "then
\n" ); document.write( "u^2 + 2u + 4 = 0
\n" ); document.write( "when using the quadratic formula, \"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\", in order to get real roots, the discriminant, \"b%5E2-4%2Aa%2Ac\", must be positive.
\n" ); document.write( "In our quadratic eqn above,
\n" ); document.write( "a = 1, b = 2, c = 4
\n" ); document.write( "bē-4ac = 4 - 4*1*4 = 4 - 16 = -12
\n" ); document.write( "i.e. the discriminant is negative, hence the eqn has no real solutions - only complex ones.
\n" ); document.write( "Since u is complex, then so also is x.\r
\n" ); document.write( "\n" ); document.write( "Fact: all polynomials of the nth degree have n solutions.
\n" ); document.write( "Fact: when complex solutions (to a polynomial) occur, they occur as conjugate pairs.
\n" ); document.write( "A cubic eqn is of the 3rd degree, hence has three solutions. It has either zero complex solutions or two complex solutions. Hence it has at least one real solution.\r
\n" ); document.write( "\n" ); document.write( "N.B. a+ib and a-ib are conjugate pairs. (change of sign for the imaginary component) \n" ); document.write( "
\n" );