44.3 Polynomial Addition

Polynomial Addition is the process of combining two or more polynomials into a single simplified polynomial by removing grouping parentheses, identifying like terms across all operands, and adding their coefficients together while carrying forward any terms that have no match.

Addition Parenthesis Removal

Procedure

Since addition does not change the sign of any term it operates on, the parentheses grouping each polynomial operand are removed without altering the sign of any term inside them.

Example

Original Term Sign Retention

Rule

Every term retains the exact sign it carried within its original polynomial operand, since removing the parentheses during addition performs no sign changes whatsoever.

Confirmation

Comparing the expression before and after parenthesis removal in the example above confirms that every term's sign, positive or negative, remains unchanged.

Matching-Term Coefficient Addition

Procedure

Terms sharing the exact same variable part are grouped together, and their coefficients are added using ordinary arithmetic to produce a single combined term.

Example

Continuing the running example, the two degree-two terms combine:

Unmatched Polynomial Term Retention

Procedure

Any term that has no matching variable part among the other operands is carried forward into the final result exactly as it originally appeared, with its coefficient and sign both unchanged.

Example

The terms $3x$ and $-5$ from the running example have no matching partner, so both are carried forward unchanged into the final sum.

Zero-Coefficient Term Removal

Procedure

If combining matching terms produces a coefficient of exactly zero, that resulting term is removed entirely from the final expression, since it contributes no value.

Example

If a polynomial contained $4x$ and another contained $-4x$, adding these produces a coefficient of zero, and the term is dropped from the final result entirely.

Polynomial Sum Standardization

Procedure

Once every matching term has been combined and every unmatched term has been carried forward, the resulting collection of terms is arranged into standard form, ordered by descending degree.

Example

Combining all steps from the running example produces the terms $3x^2$, $3x$, and $-5$, which are already arranged in the correct descending order:

Polynomial Sum Degree Determination

Procedure

The overall degree of the final summed polynomial is determined using the same degree rules established for any polynomial, selecting the greatest degree among the surviving nonzero terms.

Example

The final sum $3x^2+3x-5$ has an overall degree of $2$, the degree of its leading term.