Construct Polynomial Equations with Real Roots - Algebra-2

1. Fundamental Concepts

Definition: Polynomial equations are algebraic expressions that can be written as the sum of terms, each of which is a constant multiplied by a variable raised to a non-negative integer power.
Real Roots: Real roots are values of the variable that satisfy the polynomial equation and are real numbers.
Constructing Equations: Constructing polynomial equations with given real roots involves using the factored form of polynomials where each root corresponds to a factor.

Factored Form: $$(x - r_1)(x - r_2) \cdots (x - r_n) = 0$$

Multiplicity: If a root $r$ has multiplicity $m$, it appears $m$ times in the factored form.

Example Construction: Given roots $3, -2, 4$, construct: $(x - 3)(x + 2)(x - 4) = 0$

Problem: Construct a polynomial equation with real roots $2, -3, 5$.

Validation: The constructed equation $(x - 2)(x + 3)(x - 5) = 0$ has roots $2, -3, 5$.

Problem: Construct a polynomial equation with real roots $1, 1, -4$.

Write the factors considering the multiplicity of the root $1$: $(x - 1)^2(x + 4)$
Multiply the factors together: $(x - 1)^2(x + 4) = 0$

Validation: The constructed equation $(x - 1)^2(x + 4) = 0$ has roots $1$ (with multiplicity 2) and $-4$.

Factor Identification: Identify all given roots and their multiplicities.
Root Multiplicity Handling: For roots with multiplicity greater than one, include the corresponding factor multiple times.
Verification: Substitute the roots back into the constructed equation to verify they satisfy it.