20 Jun Answer: guess roots and factor. Quadratic equations are poly
Answer: guess roots and factor. Quadratic equations are polynomial equations whose highest power is 2 while quartic equations are polynomial equations whose highest power is 4. Any polynomial expression (whether quadratic quartic or any other power) can be factored into its roots. A root is a value which when plugged in for the variable equals zero. Example: ##x^4 + 3x^3 + x^2 – x + 2 = 2x + 4##. First rewrite as a polynomial that equals 0: ##x^4 + 3x^3 + x^2 -3x -2 = 0##. A quartic equation has at most 4 roots. It isn’t elegant but we can find them by guessing a value for ##x## and seeing whether it equals zero. 1 works here because ##1^4 + 3*1^3 + 1^2 – 3*1 – 2 = 0##. Now that we have one root 1 we know that the expression can be factored into ##(x-1)## and a cubic because plugging 1 into the ##(x-1)## term equals 0. To find the cubic we use polynomial long division: ##(x^4+3x^3+x^2-3x-2)/(x-1) = x^3 + 4x^2 + 5x + 2## Continuing we can find the other roots -1 (a double root) and -2. Or you can use the cubic or quadratic equation once you’ve reduced the polynomial. This gives the final factorization of ##(x-1)(x+1)^2(x+2) = 0##. So the solutions are ##x = 1 -1 text{or} -2##.
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