What is a polynomial problem and how can I multiply and simplify it?

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crmhaske | College Teacher | (Level 3) Associate Educator

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A polynomial is any algeberic expression that contains constants, and variables to the power of whole numbers (positive integers), and uses only the operations of addition, subtraction and multiplication (i.e., no division). 

For example this is a polynomial: `4x^3-2x+7`

And this is not a polynomial: `4x^(3/2)-2/x+7`

A polynomial problem is any question that involves the solving of a polynomial equation.  This can be anything as simple as being asked to find the polynomial equation given known data, or simplify a given algebric expression that results in a polynomial, or using a polynomial to solve a word problem, for example.

Multiplying polynomials:

When multiplying polynomials all terms in polynomial 1, must be multiplied to all terms in polynomial 2.  As in:

`(a+b-c)(-d+e+f)`

`=-ad+ae++af-bd-be-bf+cd-ce-cf`

Where a, b, c, d, e, and f are all terms in the polynomial.

Simplifying a polynomial means that you collect like terms.

Example:

Multiply `x^2-4x+4`  and `2x^2+3x-7` , and simplify the result.

`(x^2-4x+4)(2x^2+3x-7)`

`=(x^2)(2x^2)+(x^2)(3x)+(x^2)(-7)+(-4x)(2x^2)+(-4x)(3x)+(-4x)(-7)+(4)(2x^2)+(4)(3x)+(4)(-7)`

`=2x^4+3x^3-7x^2-8x^3-12x^2+28x+8x^2+12x-28` 

`=2x^4+(3x^3-8x^3)+(-7x^2-12x^2+8x^2)+(28x+12x)-28`

`=2x^4-5x^3+11x^2+40x-28`

Sources:

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