The **sum of polynomial functions** is the result of adding two or more **polynomial expressions,** which **are algebraic expressions** that include **variables** and **coefficients.**

**Polynomials** are **added** by combining like terms, terms that have the same **variables** raised to the **same powers.**

For example, to add the **polynomials** **$f(x) = 4x^2 + 3x – 5$** and** $g(x) = 2x^2 – x + 1$**, one would combine the **coefficients** of **like terms** to get the resulting **polynomial** **$h(x) = (4x^2 + 2x^2) + (3x – x) – (5 – 1)$**, which simplifies to **$h(x) = 6x^2 + 2x – 4$**.

Understanding the process of adding **polynomials** is fundamental in **algebra** and is frequently **applied** in **calculus** and higher **mathematics.**

The outcome of this **addition** yields a new polynomial whose **degree** is determined by the highest power of the **variable** present in the combined result, provided that the **sum** does not include any terms that cancel each other out completely.

This concept lays the **foundation** for more **complex operations** such as **polynomial multiplication** and **division.**

## The Sum of Polynomials

The process of adding polynomials involves **combining like terms** and aligning terms with the same **degree**.

This operation is fundamental in algebra and is performed by manipulating the **coefficients** and **variables** to achieve a simplified algebraic expression.

### Adding Like Terms

When adding polynomials, one first needs to identify the **like terms**. **Terms** are terms with the same **variables** raised to the same **power**. The actual addition involves summing the **coefficients** of these **like terms** while keeping the **variables** unchanged.

For example, consider the polynomials $3x^2 + 4x + 5$ and $5x^2 – 2x + 3$. To add these two, one would align and add the **coefficients** of the **like terms**:

$$ \begin{align*} (3x^2 &+ 4x + 5) \ (5x^2 – 2x + 3) \ (3x^2 + 5x^2) &+ (4x – 2x) + (5 + 3) \ 8x^2 &+ 2x + 8 \end{align*} $$

The result is a new polynomial in standard form, $8x^2 + 2x + 8$.

### Adding Polynomials in Standard Form

**Standard form** dictates that an algebraic expression be written with **terms** in descending order of **degree**. When **adding polynomials** presented in this form, one ensures that each **term** of similar **degree** from the polynomials is visibly lined up, which makes the addition process straightforward.

Consider adding a **linear** polynomial $2x + 3$ to a **quadratic** polynomial $x^2 – x + 1$. The addition in **standard form** appears as:

$$\begin{align*} &\phantom{0x^2 +\ }2x + 3 \& x^2 – x + 1 \ & x^2 + (2x – x) + (3 + 1) \ &x^2 + x + 4 \end{align*} $$

Thus, the sum of the **linear** and **quadratic** polynomials is the **trinomial** $x^2 + x + 4$.

## Conclusion

When one combines **polynomials**, they are effectively adding or subtracting the **polynomial expressions** to obtain a single **polynomial** as the result.

In mathematical terms, adding **polynomials** involves **summing like terms,** which are terms that have the same **variable** raised to the same power. The process emphasizes combining coefficients while keeping the **variables** and their **respective exponents** unchanged.

For instance, when **adding** two **polynomials**, **$P(x) = 3x^2 + 2x + 1$** and** $Q(x) = 5x^2 – 4x + 7$**, one would align the like terms and sum the coefficients to get the **resulting polynomial $R(x) = (3x^2 + 5x^2) + (2x – 4x) + (1 + 7)$**.

Thus, the resulting **polynomial $R(x)$** after performing the addition would be **$8x^2 – 2x + 8$**.

Subtraction is treated similarly, where one **subtracts** the **coefficients** of like terms of the **subtrahend polynomial** from the corresponding terms of the **minuend polynomial.**

In both **operations,** care is taken to ensure that the integrity of the original **exponents** and **variables** is maintained.

These **operations** are fundamental in algebra and are **integral** for solving more **complex algebraic problems.** Mastery of **polynomial addition** and **subtraction** is essential for progress in understanding **algebraic equations** and **functions.**