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     Polynomial Operations Calculator

Instructions: Enter two polynomials and specify the operation you want to conduct among sum, subtraction or product, and the solver will show you step-by-step how to get the result. Type the polynomials like '3x^2 + 2x + 3'

Type the 1st polynomial \(p_1(x)\):    

Type the 2nd polynomial \(p_2(x)\):   

Select the operation you want to perform:

\(p_1(x) + p_2(x)\)
\(p_1(x) - p_2(x)\)
\(p_1(x) \cdot p_2(x)\)

More About Polynomial Operations

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

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

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

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

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