Algebra: Polynomials, Matrices, and Linear Systems
Polynomials and equations
A polynomial’s graph meets the x-axis at its real zeros; for y=ax²+bx+c, the parabola opens upward when a>0 and downward when a<0, with vertex x=−b/2a. The quadratic formula gives roots [−b±√(b²−4ac)]/2a; the discriminant b²−4ac decides two, one, or no real roots. For two polynomials, GCD is the product of common factors with least powers, LCM uses greatest powers; for monic polynomials, product=GCD×LCM. Square root of a polynomial is found by long division, pairing terms from the highest degree.
Matrices
For A=[[a,b],[c,d]], det A=ad−bc. A is singular if det A=0. Matrix multiplication is not commutative; AB is defined only when columns of A equal rows of B. A system a₁x+b₁y=c₁, a₂x+b₂y=c₂ can be solved by elimination or by determinants: x=(c₁b₂−c₂b₁)/(a₁b₂−a₂b₁), y=(a₁c₂−a₂c₁)/(a₁b₂−a₂b₁), provided the denominator is nonzero.
Worked example
Solve x²−5x+6=0. Factor: (x−2)(x−3)=0, so x=2 or 3. The graph y=x²−5x+6 cuts the x-axis at 2 and 3, has vertex x=2.5, and opens upward. Sum of roots=5, product=6, matching −b/a and c/a.
Traps
Dropping the minus sign in −b; reversing the determinant ad−bc; using LCM where GCD is asked; and forgetting that a zero determinant means no unique solution, not necessarily no solution.