POLYNOMIAL EXPRESSIONS PART 1 A polynomial is an expression that is a sum of one or more terms. Each term consists of one or more variables multiplied by a coefficient. Coefficients can be negative, so don t be surprised if you see a minus sign in a polynomial that just means there s a term with a negative coefficient. Here are some examples of polynomials: 5x 2 3x 3 + 2xy y 3x 2 + 6x 7 Polynomials are classified by the number of terms they have when they are expressed in their simplest form. A monomial has one term, a binomial has two terms and a trinomial has three. 3x + 2 is a binomial, because it has two terms. x 2 4x + 2 is a trinomial, because it has three terms. Polynomials are also classified by their degree. The degree of a term is the sum of the exponents of its variables. The degree of a polynomial is the same as its the highest degree term. For example, x 2 + 3 is a second-degree polynomial because its highest exponent is 2. The expression x 3 + x 2 + 2x + 1 is a third-degree polynomial because its highest exponent is 3. MATH 588
Certain polynomials have special names determined by their degree: Degree Name Example 0 Constant 5 1 Linear x + 7 2 Quadratic x 2 + 9 3 Cubic 2x 3 + 19x 2 6x + 13 Just like regular numbers, polynomials can be added, subtracted, multiplied, and divided. Next up, we ll cover how to add, subtract, and multiply polynomials, as well as some techniques for basic division. ADDING POLYNOMIALS To add two polynomials, you need to combine the like terms. Like terms have the same variables raised to the same powers. So 5x 3 y 2 z and 7x 3 y 2 z are like terms because each term has x cubed, y squared, and z. However, 5x 3 y 2 z and 7xyz are not like terms, because although they have the same variables, the variables are not raised to the same powers. Let s say you want to find the sum of 3m 2 + 2m + 6 and m 9. You can join them with a plus sign: 3m 2 + 2m + 6 + m 9 Then, put like terms next to each other. Remember to pay attention to the signs! 3m 2 + 2m + m + 6 9 Finally, add and subtract the like terms, including the constants: 3m 2 + 3m 3 MATH 589
SUBTRACTING POLYNOMIALS Subtracting polynomials is very similar to adding them: join the expressions and combine like terms. However, with subtraction, you first have to take care of the signs. What is the value of 4p 3 + 6p 2 8p + 11 minus 3p 3 2p 2 + 12p 3? Just like with addition, you ll need to join the terms. With subtraction, however, you need to put the second term in parentheses: 4p 3 + 6p 2 8p + 11 (3p 3 2p 2 + 12p 3) Then, distribute the negative sign across the parentheses: 4p 3 + 6p 2 8p + 11 3p 3 + 2p 2 12p + 3 Now you re ready to combine like terms for your result: p 3 + 8p 2 20p + 14 MULTIPLYING POLYNOMIALS To multiply two monomials, use exponent rules. 5x 3 y 5 z 2 2x 6 y 8 z Remember that when you multiply two expressions with the same base, you can add their exponents. Don t forget to multiply the coefficients! 5x 3 y 5 z 2 2x 6 y 8 z = 10x 3 + 6 y 5 + 8 z 2 + 1 MATH 590
Once you do the arithmetic, you re left with: 10x 9 y 13 z 3 When you multiply polynomials with more terms, you will need to use the Distributive Property, which we talked about in Section 4, Part 1. Take a moment to review the Distributive Property, and then look at this example: 2x(x + 3) Using the Distributive Property, you can rewrite the expression like this: 2x x + 2x 3 Then, simplify to get your solution: 2x 2 + 6x When you re multiplying more than one polynomial with multiple terms, the idea is the same: use the Distributive Property and simplify. You just have to make sure you ve multiplied every term in one polynomial by every term in the other. Luckily, for multiplying two binomials, there s an easy way to keep everything straight. The FOIL method tells you to multiply the First terms, the Outer terms, the Inner terms, and the Last terms. Always remember to combine like terms when you ve finished. (x + 3)(2x + 5) You need to multiply both terms in the first binomial by both terms in the second, like this: (x + 3)(2x + 5) MATH 591
The FOIL method makes this simple. Multiply together the first terms in the parentheses (x and 2x), then the outer terms (x and 5), then the inner terms (3 and 2x), and finally the last terms (3 and 5): (x 2x) + (x 5) + (3 2x) + (3 5) Then, simplify and combine like terms: 2x 2 + 5x + 6x + 15 2x 2 + 11x + 15 DIVIDING POLYNOMIALS To divide polynomials, you can use the same techniques you learned for factoring expressions. 2x 3 + 4x 2 6x 2x This operation is asking you to divide each term of the polynomial by 2x. Remember that dividing two expressions with the same base means you can divide the coefficients and subtract the exponents: 2 2 x3 1 + 4 2 x2 1 6 2 x1 1 Carry out that arithmetic, and simplify the coefficients where you can: x 2 + 2x 3 Don t worry if the number you re dividing the coefficients by isn t a common factor. It s fine to leave coefficients as fractions in lowest terms. MATH 592
PART 1 PRACTICE: POLYNOMIALS 1. Add x 3 + 2x 6 and 4x 3 5x + 9. 2. Find the sum of m 4 + 4m 3 m 2 + 17m 2 and 3m 4 m 3 2m 2 + 3m 8. 3. Find the value of 3x + 16 minus x + 8. 4. Subtract 10x 10 from 4x + 7. 5. 12m 2 n 7 5m 10 n 3 = 6. (x 7)(x + 3) = 7. (2x + 5)(x 11) = 8. 5x 2 + 10x 15 = 5 9. What is 100x 4 + 44x 2 divided by 4x 2? 10. 3x4 + 36x 3 12x 2 3x = MATH 593
ANSWER KEY: POLYNOMIALS 1. 5x 3 3x + 3 2. 4m 4 + 3m 3 3m 2 + 20m 10 3. 2x + 8 4. 6x + 17 5. 60m 12 n 10 6. x 2 4x 21 7. 2x 2 17x 55 8. x 2 + 2x 3 9. 25x 2 + 11 10. x 3 12x 2 + 4x MATH 594