Overview: What Are Polynomials?
Polynomials are a special type of mathematical expression created by adding and subtracting numbers and variables. They can be added, subtracted, multiplied, and many other operations. Expressions such as the number 2, 4a, b2, and 3x3 are all monomials that can be combined into number sentences as polynomials.
Variables, Constants, and Coefficients
Algebra is based upon the interaction of constants and variables. Numbers such as 2, 4, and 5 always mean the same thing and can be combined in familiar operations. They are the basis of math facts. In polynomials, they are constants. Variables in math are letters that stand for other numbers, the a, b, c, and x, y, z of algebra. They can be replaced by many different values depending on the solution of the problem. Coefficients are special constants that combine with variables. In the monomial 3x3, 3 is the coefficient.
Adding and Subtracting Polynomials
In order to add or subtract polynomials, group like terms together. For example, when adding (4a +6b) + (2a-3b), first group the (4a +2a) and the (6b-3b) together. Then use the distributive property, to solve (4+2)a + (6-3)b, or 6a +3b. Subtraction is the inverse of addition. Positive integers become negative , and negative become positive. Suppose the sentences are (r+ 8s)-(r-5s). Grouping the rs together results in r-r =0 and 8s +5s = 13s.
Multiplying Polynomials
When multiplying polynomials, use the distributive property, as well as exponents. For example, if a rectangle has one side 3x and another side (7x-9), to find the area, multiply the length by the width to find the area, or 3x(7x -9) = 3x(7x)+ 3x(-9) = 21x2-27x.
Multiplying two polynomials involves using the distributive property twice. For example, multiply (x+2)(x-5) to equal (x+2)x + (x+2)(-5) or x2+ 2x -5x -10. A positive number multiplied by a negative number will result in a negative number. Grouping like terms will result in x2+ (2x-5x) -10 or x2-3x-10.
FOILed Again
In order to help remember the order of multiplication of two polynomials, use the word FOIL to remember the first terms are combined first, then the outer terms, then the inner terms, then the last terms. For example, when multiplying (2x +5)(3x+4), first multiply 2x(3x) or 6x2, then 2x (4) or 8x, then 5(3x) or 15x, then 5(4) or 20. Adding like terms together, 6x2 + (8x+ 15x), or 23x + 20.
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