Polynomials and Linear Equations in Two Variables
This section covers Polynomials and Linear Equations in Two Variables. The exam tests zeros and coefficient relationships, the polynomial division algorithm, and solving or classifying pairs of linear equations.
Polynomial rules
- A zero of p(x) is a value c with p(c) = 0.
- For ax² + bx + c, the sum of zeros is -b/a and the product is c/a.
- Division algorithm: p(x) = g(x)q(x) + r(x), where the degree of r is less than the degree of g.
Linear equations
Write both equations as a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 before comparing ratios.
- a₁/a₂ ≠ b₁/b₂: one solution, intersecting lines.
- a₁/a₂ = b₁/b₂ = c₁/c₂: infinitely many solutions, coincident lines.
- a₁/a₂ = b₁/b₂ ≠ c₁/c₂: no solution, parallel lines. Substitution or elimination gives an exact solution; a graphical solution should be checked in both equations.
Worked example
Solve 2x + 3y = 12 and x - y = 1. From the second equation, x = y + 1. Substitute: 2(y + 1) + 3y = 12, so 5y + 2 = 12 and y = 2. Then x = 2 + 1 = 3. Check: 2(3) + 3(2) = 12 and 3 - 2 = 1.
Traps
A pair can look inconsistent if the equations are not in standard form; rearrange first. Watch signs when subtracting equations. In polynomial division, include missing powers with zero coefficients, and stop when the remainder’s degree is lower than the divisor’s.