More About this Polynomial Operations Calculator

Polynomials operations are operations that can be conducted among polynomials. Polynomials can be added, subtracted, multiplied and divided, regardless the order of the polynomial.

For example, we can add the polynomials \(p_1(x) = x + 3\) and \(p_2(x) = 2x - 1\) as follows

\[p_1(x) + p_2(x) \] \[= (x+3) + (2x - 1)\] \[= x+3 + 2x - 1\] \[= x + 2x + 3 - 1\] \[= 3x + 2\]

How to use this Polynomial operations calculator with steps

The procedure is simple: Just put the polynomials together and group by exponent and add the terms up. The same procedure is applied when adding polynomials of different order.

For example, let us add \(p_1(x) = x^2+3\) and \(p_2(x) = 2x - 1\) as follows

\[p_1(x) + p_2(x) \] \[= (x^2+3) + (2x - 1)\] \[= x^2+3 + 2x - 1\] \[= x^2 + 2x + 3 - 1\] \[= x^2 + 2x + 2\]

Almost exactly the same methodology is applied when we subtract polynomials, as indeed, subtracting \(p_2(x)\) from \(p_1(x)\) is the same as taking \(p_2(x)\), multiplying each coefficient by \(-1\) and then add this resulting polynomial to \(p_1(x)\)

For the multiplication of polynomials, things can get a bit messier because we need to cross multiply all the terms in one polynomials with the terms of all other polynomials.

For example, let \(p_1(x) = x^2+3\) and \(p_2(x) = 2x - 1\), let's compute the multiplication

\[p_1(x) \cdot p_2(x) \] \[= (x^2+3) \cdot (2x - 1)\] \[= (x^2)\cdot (2x)+ (x^2)\cdot (-1) + (3)\cdot (2x)+ (3)\cdot (-1)\] \[= 2x^3 - x^2 + 6x - 13\]

Polynomial function calculator graph

There are lots of things you can do with polynomials. On the one hand, you could plot a polynomial to get an idea of how the polynomial behaves.

Then, you can also compute the polynomial roots, using a systematic procedure to attempt to find all the roots (real and complex), which is not always possible using elementary methods.

Then, you can also use a tool like the Descartes Rule of Signs to calculate the number of positive and negative roots, based on the number of sign changes between consecutive polynomial coefficients.

Aside from this polynomials calculator, you can choose among our selection of algebra calculators .