Add Subtract Polynomials Calculator - Sum and Difference

An add subtract polynomials calculator is a single tool that takes two single-variable polynomials, P(x) and Q(x), and returns either their sum P(x) + Q(x) or their difference P(x) - Q(x) in standard form. You pick the operation, type the coefficients of P(x) and Q(x) from the highest power of x down to the constant term, and the result panel shows the standard-form polynomial, its highest degree, and the count of non-zero terms.

• • Homework and textbook check: Verify the result of a manual polynomial sum or difference without redoing the like-term arithmetic on scratch paper.
• • Combine two fitted curves: Add or subtract two polynomial expressions when modeling data, such as stacking a trend curve with a seasonal correction.
• • Simplify a symbolic expression: Reduce an expression that is a sum or difference of two polynomials to standard form before factoring, graphing, or substitution.
• • Prepare inputs for follow-up tools: Generate the standard-form sum or difference that will be the dividend, divisor, or argument for the related polynomial tools.

Adding and subtracting polynomials is the algebra operation that makes almost every other single-variable tool work. The sum or difference inherits the same variable and keeps the same exponents, so only the coefficients change.

When the result of the sum or difference needs to be divided by another polynomial, our Polynomial Division Calculator applies the same coefficient-by-coefficient form to long division.

The calculator lines up the two polynomials by power of x, then adds or subtracts the coefficient of each power. Powers that do not appear in one polynomial are treated as zero, and the term carries through unchanged.

• P(x): The first polynomial, with coefficients a4, a3, a2, a1, a0 for x^4 down to the constant term.
• Q(x): The second polynomial, with coefficients b4, b3, b2, b1, b0 for the same powers of x.
• a_i, b_i: The coefficient of x^i in P(x) and Q(x) respectively. Zero when the polynomial has no term of that power.
• sign: Equals +1 for add and -1 for subtract. The same coefficient lists are combined twice to produce both the chosen result and the other operation.

The same coefficient lists are combined twice, once with sign +1 and once with sign -1. Switch the operation selector to swap which of the two rows is highlighted, without retyping any of the eleven numeric inputs. Zero terms are removed so the output matches the form students write on a homework page.

According to Omni Calculator, the rule for adding or subtracting polynomials is to combine only like terms by adding or subtracting the coefficients of each power of x, and to carry the unmatched terms through unchanged.

If the result of the sum or difference is a degree 2 or degree 3 polynomial that you want to factor next, our Polynomial Factorization Calculator factors quadratics and cubics in standard form.

Four small ideas make the calculator's work predictable and explain why the result can drop the x^2 term in one example and grow to five terms in another.

These four ideas are the entire rule set. The rest of the calculator is just doing the same addition or subtraction at every power of x and presenting the surviving terms in the conventional order.

Synthetic division uses the same coefficient rules when the divisor is linear, so a simplified sum can be handed to our Synthetic Division Calculator to recover the quotient and remainder.

Pick the operation, type the two polynomials as coefficients from the highest power down to the constant term, and read the result on the right.

1. 1 Choose the operation: Pick Add (P(x) + Q(x)) or Subtract (P(x) - Q(x)) in the Operation field.
2. 2 Enter the coefficients of P(x): Type the coefficient of x^4, x^3, x^2, x, and the constant term in the five P(x) fields. Use 0 for any missing term.
3. 3 Enter the coefficients of Q(x): Type the coefficient of x^4, x^3, x^2, x, and the constant term in the five Q(x) fields, using 0 for any missing term.
4. 4 Read the result polynomial: Look at the highlighted Result Polynomial row for the standard-form sum or difference.
5. 5 Check the highest degree and term count: Use the Result Highest Degree and Non-Zero Terms rows to confirm the structure.
6. 6 Read the other operation: The Other Operation row shows the polynomial from the opposite sign without retyping any coefficients.

Example: a student is asked to add 2x^3 + 3x - 1 and x^3 - 4x + 5. They pick Add in the Operation field, type 2 in P(x) x^3, 0 in P(x) x^2, 3 in P(x) x, -1 in P(x) constant, 1 in Q(x) x^3, 0 in Q(x) x^2, -4 in Q(x) x, and 5 in Q(x) constant. They read 3x^3 - x + 4 in the highlighted Result Polynomial row, with the Other Operation row showing x^3 + 7x - 6.

Once the sum or difference is in standard form, our Polynomial Graphing Calculator plots the result on the same axes as P(x) and Q(x) so the visual effect of the addition or subtraction is visible.

The bookkeeping is exactly the kind of arithmetic that breaks in long homework sets or multi-step modeling problems.

• • One pass for all five like-term combinations: Type the five coefficients of P(x) and the five of Q(x) once, and the calculator adds or subtracts every power of x at the same time.
• • Sum and difference from one set of inputs: The other-operation row gives the polynomial from the opposite sign using the same coefficients.
• • Standard form without sign mistakes: The result string is built from the highest non-zero power down to the constant term with a plus or minus between every term.
• • Works with decimals and negatives: Coefficient inputs such as 0.5, -1.25, and -3 are kept as exact values, then formatted with trailing zeros trimmed.
• • Highest degree and term count visible together: The result panel reports the highest degree and the non-zero term count next to the polynomial string.

The biggest practical benefit is that the operation is explicit. A student can show the work by reading the result row, while a tutor can scan the coefficient lists to see which input produced the cancellation.

If the result turns out to be a quadratic with three non-zero terms, our Factoring Trinomials Calculator factors the trinomial so the sum or difference can be written in factored form.

The arithmetic is fixed, but the choice of operation and the value of the coefficients change the shape of the result.

• • The calculator accepts two polynomials in one variable, x. Polynomials in two or more variables such as x^2 + xy + y^2 are not supported.
• • The input panel handles polynomials up to degree 4 in each of P(x) and Q(x). Higher-degree inputs have to be split, the relevant part worked on here, and the higher-degree remainder handled separately.

For most algebra work the limitation is a non-issue, because the common textbook case is a single-variable polynomial of degree 2, 3, or 4. When the result is a quadratic, the factoring trinomials calculator picks up the next step. When the result has to be graphed, the polynomial graphing calculator plots it.

According to Wolfram MathWorld, a polynomial is a sum of scalar coefficients times non-negative integer powers of an indeterminate, and the standard form lists those powers from the highest degree down to the constant term.

According to Paul's Online Math Notes, adding or subtracting polynomials comes down to combining like terms by adding or subtracting the numerical coefficients of each power of x.

When the result of the sum or difference is a degree 2 polynomial with three non-zero terms, our Quadratic Formula Calculator solves the equation form of that quadratic and returns its real roots.