Adding and Subtracting Polynomials Calculator — Sum & Difference Step by Step

Adding and Subtracting Polynomials: Sum and Difference Made Simple

Welcome to CalcuGo's Adding and Subtracting Polynomials Calculator, where we'll learn to find the sum and difference of polynomials. In fact, the operation is fairly simple — much simpler than polynomial multiplication. Nevertheless, let's go through it slowly, step by step, to get a clear understanding of the process. Once we know how to add and how to subtract polynomials, we'll walk through a nice example at the end. If you'd rather multiply, try our multiplying polynomials calculator instead.

Polynomials, Monomials, Binomials

Ever since we went past counting apples and candies, we've been handling polynomials without even knowing it. A polynomial is an algebraic expression (not an equation!) in which the variables appear only in non-negative integer powers. In other words, it cannot contain radicals, logarithms, or trigonometric functions with letters inside them. A few examples of polynomials:

• x + 2y
• a² + 2ab + b
• πr²
• n³ − 0.7n + ⅜

Binomials and monomials are special cases of polynomials: they are those with two and one terms, respectively (the prefixes bi- and mono- mean double and single). There are also trinomials, which have three terms, appearing mostly in quadratic equations.

Our adding and subtracting polynomials calculator deals only with expressions that have one variable (here it's x) and at most power 6. Such objects are the most common in algebra and are enough to explain the concept.

How to Add Polynomials?

Just like in arithmetic, finding the sum in this new setting is a fairly easy task. The basic rule on how to add polynomials is: combine only similar terms. More precisely, we can only add monomials with the same variables and variable powers (but possibly different constants in front). For instance, we can add 3xy² to −0.5xy², but not to 2x²y², since the power of x is different.

Suppose we have two polynomials P(x) = a₅x⁵ + a₄x⁴ + a₃x³ + a₂x² + a₁x + a₀ and Q(x) = b₄x⁴ + b₃x³ + b₂x² + b₁x + b₀. We can add only terms that have the same power of the variable — combining means simply adding the numbers in front:

P(x) + Q(x) = a₅x⁵ + (a₄ + b₄)x⁴ + (a₃ + b₃)x³ + (a₂ + b₂)x² + (a₁ + b₁)x + (a₀ + b₀)

Observe how the first summand has only a₅ in front, since there was no x⁵ term in Q(x). And that's all there is to it — easy, wasn't it?

How to Subtract Polynomials?

Subtracting polynomials is as easy as adding them; we simply change sums to differences throughout. To subtract P(x) and Q(x), the rule is the same: we're only allowed to combine alike terms, and combining now means finding the difference of the coefficients:

P(x) − Q(x) = a₅x⁵ + (a₄ − b₄)x⁴ + (a₃ − b₃)x³ + (a₂ − b₂)x² + (a₁ − b₁)x + (a₀ − b₀)

The formula looks the same, but we need to be careful with the signs. Addition is commutative, so the order of the summands didn't matter. Subtraction is not commutative, so the minus signs must appear only in front of the b terms — the coefficients of the subtrahend.

Example: Using the Adding and Subtracting Polynomials Calculator

Let's try both options with:

• P(x) = 4x⁴ − x³ + 5x + 1
• Q(x) = x⁵ + 4x⁴ − 7x³ − 3x² + x + 12

Take special care of a few things: a coefficient of 1 is not written (so x⁵ means b₅ = 1), a missing term means its coefficient is 0 (P(x) has no x², so a₂ = 0), and minus signs carry into the coefficient (−x³ means a₃ = −1).

The sum:

P(x) + Q(x) = x⁵ + 8x⁴ − 8x³ − 3x² + 6x + 13

The difference:

P(x) − Q(x) = −x⁵ + 6x³ + 3x² + 4x − 11

Voilà! It wasn't so bad, was it?

How to Use This Calculator

1. Choose whether you want to add or subtract at the top of the form.
2. Type your first polynomial P(x) into the first field (e.g., 4x^4 - x^3 + 5x + 1).
3. Type your second polynomial Q(x) into the second field.
4. Click Calculate.
5. View the simplified result and the step-by-step, power-by-power combination table.

Input Format Guide

• Use ^ for exponents: x^3 means x³
• Use + and - between terms
• Coefficients can be integers or decimals: 2.5x^2
• A bare x means 1x¹; a bare number (e.g., 7) is a constant term
• Negative coefficients: -3x^2 + x - 5