To better understand the rules on how to add and subtract polynomials, let us first define polynomials and other concepts associated with it such as term, similar terms, and degree of polynomials. Polynomials are those algebraic expressions comprised of a constant and variable term(s) having only whole number exponents joined by addition, subtraction or multiplication and there must be no variable in the denominator. A term is a distinct part of an algebraic expression together with the sign prior to it. For example, in the expression 3x – 5xy + 10, the parts 3x, -5xy, 10 are each called term. Similar terms are those terms that have the same variables and have the same degree. In the polynomial, 8x + 9y – 15x - 5, the parts 8x and 15x are similar terms. The degree of polynomials in one variable is the exponent of the term with the highest power and in more than one variable, is the highest sum of the exponents of the variables. In the expression \(4x^3+5x^2-3x+4\), the term \(4x^3\) has the highest power so the degree of this polynomial is 3. In \(4x^3y^2+5x^2y-3xy+7\), the term \(4x^3y^2\) has the highest sum of exponents, that is, \(3+2=5\) so the degree of this polynomial is 5.
Adding and Subtracting Polynomials can be done in two ways; either horizontally or vertically.
The following are the rules in adding and subtracting polynomials:
1. To add two or more polynomials, group similar terms and find their sum. Arrange the final answer from the highest degree to the lowest.
1.1. In horizontal addition, remove the parenthesis without changing any sign and combine similar terms by adding or subtracting their numerical coefficients.
1.1.1 Add \(2x+5y-12\) and \(5x-2y+10\).
1.1.2 Simplify: \(\left(x^2+xy-y^2\right)+ \left(x^3-x^2+2xy-5\right)\).
1.2 In vertical addition, put similar terms in one column then add.
1.2.1 Add \(\left(x^2+2xy-2y^2\right)+\left(2x^2-5xy+4y^2\right)\).
2. To subtract one polynomial from another, change the sign of each term of the subtrahend and proceed to addition.
2.1 Horizontal Subtraction
2.2 Vertical Subtraction
2.2.1 Subtract \(x+4y+2z\) from \(5x+3y-7z\).
- \(\left(2x-3y+2\right)- \left(x-y-1\right)\)
- \(\left(2x+3y+2\right)+ \left(x-y-1\right)\)
- \(\left(2x+3y+2\right)+ \left(x-y-1\right)- \left(5x-2y+2\right)\)