what is inreducible quadratic factor?
I suspect you mean irreducible. It is quadratic expression of the form ax^2 + bx + c that cannot be factored into monomials (a'x + b')(c'x + d') with real constants. This happens if b^2 < 4 ac
See
http://www.sosmath.com/algebra/factor/fac04/fac04.html
for further information
I need help with simplify fraction
An irreducible quadratic factor refers to a quadratic polynomial that cannot be factored further over the set of rational numbers. In other words, it cannot be factored into any combination of linear or quadratic polynomials with rational coefficients.
To determine whether a quadratic polynomial is irreducible, you can follow these steps:
Step 1: Express the quadratic polynomial in its standard form.
Step 2: Check if the polynomial can be factored using integer coefficients. If it can, then it is not irreducible.
Step 3: If the polynomial cannot be factored using integer coefficients, you can use the quadratic formula to find its roots.
Step 4: If the roots of the quadratic are not rational, then the polynomial is irreducible. However, if the roots are rational, it may still be reducible.
Step 5: Use long division or synthetic division to perform polynomial division. Divide the original polynomial by the possible factors that could be formed by the rational roots. If the division does not yield zero remainder, then the polynomial is irreducible. If it does, then the polynomial can be factored further.
It's important to note that irreducible quadratic factors can only be found over a particular set of numbers. For example, over the set of real numbers, some quadratic polynomials may appear to be irreducible, but they can actually be factored into linear factors over the set of complex numbers.