Adding and Subtracting: Polynomials
Adding and subtracting polynomials is a fundamental concept in algebra that involves the manipulation of algebraic expressions consisting of terms with variables and constants. Polynomials can be categorized based on their number of terms: a monomial has one term, a binomial has two, and a trinomial has three. When adding or subtracting polynomials, particularly trinomials, it's essential to combine only like terms, which are terms that share the same exponent.
For instance, to add two trinomials, you first write out the entire expression, remove parentheses, and then combine coefficients of like terms. Similarly, when subtracting, it is crucial to distribute the negative sign across all terms of the polynomial being subtracted to ensure accurate results. The resulting polynomial after these operations may end up having more than three terms, transforming it beyond its original trinomial form. Understanding these processes allows for more complex algebraic manipulations and is critical for further studies in mathematics. This knowledge can be applied across various fields, reflecting the versatility and importance of polynomials in mathematical problem-solving.
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Adding and Subtracting: Polynomials
Polynomials are algebraic expressions that involve sums of terms, which consist of constants and variables with non-negative exponents. Polynomial expressions that contain only one variable are sometimes called univariate polynomials or polynomials in one variable.
Polynomial terms in one variable can be written in the form axn where ais called the coefficient and can be any real number, x is a variable, and the exponent n is a non-negative integer. Examples of polynomials in one variable include 3x4, x2 + x + 111/2, −3x3 − 2x2 − x, x10 − 29x3 − 31x + 5/23, − x2/7, and so on.
The degree or order of a polynomial in one variable is the largest exponent in the entire polynomial expression. The degrees or orders of polynomials in the examples above can be seen in Table 1 below:
Polynomials can also consist of more than one variable. They can contain terms written in the form axmyn where the coefficient a is any real number, x and y are variables, and the exponents m and n are any non-negative integers.
Addition and Subtraction of Trinomials
A polynomial expression with exactly three terms is called a trinomial. (A binomial contains exactly two terms and a monomial has only one term.) When adding or subtracting trinomials, remember to only add or subtract the coefficients of terms with the same exponents. Below are examples of trinomial addition and subtraction problems.
To solve this problem, first write down the algebraic expression for the operation. In this case, the problem requires the addition of a trinomial to another trinomial.
The parentheses can be removed to form the expression 10x2 + 6x − 2 − x3 + 7x + 19. Then, the like terms (terms with the same exponents) are combined to form −x3 + 10x2 + (6 + 7)x + (−2 + 19). This can be simplified as −x3 + 10x2 + 13x + 17.
Therefore, the sum of the two trinomials is −x3 + 10x2 + 13x + 17.
Again, as with addition, first write down the algebraic expression for the operation. This time, the question is asking for the difference between 9x4 − 3x2 + x/2 and x5 + 4x3 − 20, which can be written as:
When removing the parentheses, make sure that the negative sign is distributed to all the terms of the second polynomial. Thus, the new expression is 9x4 − 3x2 + x/2 − x5 − 4x3 + 20. Since the expression does not contain any like terms, no other operations can be performed.
Therefore, the difference between the two trinomials is −x5 + 9x4 − 4x3 - 3x2 + x/2 + 20.
Note that the answers are no longer trinomials because they have more than three terms. This can happen during the addition or subtraction of trinomials with different orders/degrees.
Bibliography
Aufmann, Richard, Vernon Barker, and Richard Nation. College Algebra and Trigonometry. 7th ed. Belmont, CA: Cengage, 2011.
Montgomery, Douglas C., Elizabeth A. Peck, and G. Geoffrey Vining. Introduction to Linear Regression Analysis. Hoboken, NJ: Wiley, 2012.
Stahl, Saul. Introductory Modern Algebra: A Historical Approach. Hoboken, NJ: Wiley, 2013.
Young, Cynthia Y. Algebra and Trigonometry, 3rd ed. Hoboken, NJ: Wiley, 2013.