Addition of Polynomials

Definition

The addition of polynomials involves adding corresponding terms from each polynomial. Corresponding terms are those that have the same variable raised to the same power. The sum is obtained by adding the coefficients of these corresponding terms and keeping the variable part unchanged.

Arrange Polynomials: Write each polynomial in a standard form (i.e., in descending order of the degrees of its terms).
Identify Like Terms: Identify the terms in both polynomials that have the same variables raised to the same powers.
Add the Coefficients: Add the coefficients of the like terms.
Write the Result: Combine the results to form the new polynomial.

Examples

Example 1: Adding Two Polynomials

Add the polynomials $3x^2 + 2x + 5$ and $x^2 + 4x + 3$.

First Polynomials: $3x^2 + 2x + 5$

Second Polynomial: $x^2 + 4x + 3$

The sum is:

\[ (3x^2 + 2x + 5) + (x^2 + 4x + 3) \] $$ (3x^2 + x^2) + (2x + 4x) + (5 + 3) $$ $$ = 4x^2 + 6x + 8 $$

Example 2: Adding Polynomials with Different Degrees

Add the polynomials $4x^3 + 2x^2 - x + 7$ and $-x^3 + 3x^2 + 2x - 5$.

First Polynomials: $4x^3 + 2x^2 - x + 7$

Second Polynomial: $-x^3 + 3x^2 + 2x - 5$

The sum is:

\[ (4x^3 + 2x^2 - x + 7) + ( - x^3 + 3x^2 + 2x - 5 ) \] $$ (4x^3 - x^3) + (2x^2 + 3x^2) + ( -x + 2x) + (7 - 5) $$ $$ = 3x^3 + 5x^2 + x + 2 $$

Example 3: Adding Polynomials with Missing Terms

Add the polynomials $5x^4 + 3x^2 + 6$ and $2x^4 + x^3 - 4x$.

First Polynomials: $5x^4 + 3x^2 + 6$

Second Polynomial: $2x^4 + x^3 - 4x$

The sum is:

\[ (5x^4 + 3x^2 + 6) + (2x^4 + x^3 - 4x) \] $$ (5x^4 + 2x^4) + (x^3) + (-4x) + (6) $$ $$ = 7x^4 + x^3 + 3x^2 - 4x + 6 $$

Example 4: Adding Polynomials with Negative Coefficients

Add the polynomials $-3x^2 + 4x - 2$ and $-x^2 - 3x + 5$.

First Polynomials: $-3x^2 + 4x - 2$

Second Polynomial: $-x^2 - 3x + 5$

The sum is:

\[ (-3x^2 + 4x - 2) + ( - x^2 - 3x + 5 ) \] $$ (-3x^2 - x^2) + (4x - 3x) + (-2 + 5) $$ $$ = -4x^2 + x + 3 $$