Polynomials & Linear Pairs

Explore polynomials, understand their zeroes geometrically, and visualize the solutions to pairs of linear equations.

Polynomials & Degrees

Change coefficients to build a polynomial. The highest power with a non-zero coefficient gives the degree.

Degree: 2 (Quadratic)

Coeff. of x³ (a)

Coeff. of x² (b)

Coeff. of x (c)

-3

Constant (d)

-4

Linear PolynomialDegree 1. Highest power of x is 1. (e.g., ax + b)

Quadratic PolynomialDegree 2. Highest power of x is 2. (e.g., ax² + bx + c)

Cubic PolynomialDegree 3. Highest power of x is 3. (e.g., ax³ + bx² + cx + d)

The zeroes of a polynomial are the x-coordinates where the graph intersects the x-axis.

y = x² - 3x - 4

Zeroes: x = -1, 4

a (x²)

b (x)

-3

c (constant)

-4

Two ZeroesThe parabola intersects the x-axis at two distinct points.

One ZeroThe parabola just touches the x-axis at exactly one point.

No ZeroesThe parabola is completely above or completely below the x-axis.

A pair of linear equations in two variables forms two straight lines. Their relationship determines the solution.

x - 2y = 0

3x + 4y = 20

Intersecting Lines (1 Solution)

a₁

b₁

-2

c₁

a₂

b₂

c₂

IntersectingLines cross at a single point. Unique solution (consistent). a₁/a₂ ≠ b₁/b₂

ParallelLines never cross. No solution (inconsistent). a₁/a₂ = b₁/b₂ ≠ c₁/c₂

CoincidentLines overlap exactly. Infinitely many solutions. a₁/a₂ = b₁/b₂ = c₁/c₂

Solve pairs of equations using Substitution or Elimination.

(1) 7x - 15y = 2

(2) x + 2y = 3

Solution: x = 49/29, y = 19/29