Polynomials: Add and Subtract

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1 GSE Advanced Algebra Operations with Polynomials Polynomials: Add and Subtract Let's do a quick review on what polynomials are and the types of polynomials. A monomial is an algebraic expression that is a constant, a variable, or a product of a constant and one or more variables. The number 2 is considered a monomial because it can be written as which is 2. It is also called a constant because the degree is 0. Every number by itself is a constant. Some examples of monomials are 2,x,3x,5xy,, and x/3. A polynomial is the sum or difference of two or more monomials. Some examples of polynomials are x+2, +y 1, and 4ab 3b. Some polynomials have special names. A binomial is the sum or difference of two monomials. An easy way to remember this is 'bi' means 2, so a bicycle has two wheels. For example, x + 2 is a binomial. A trinomial is the sum or difference of three monomials. An easy way to remember this is 'tri' means 3, so a triangle has three sides. For example, 4x² 7x + 3y is a trinomial. There are no special names for polynomials with more than 3 terms. The exponents of the variables must be positive integers to be a polynomial. Since an expression with the variable in the denominator has a negative exponent it would not be a polynomial. To find the degree of the polynomial, you must find the degree of each monomial. In other words, add up the exponents of each term. The degree is determined by the exponent or sum of exponents that has the greatest value within the polynomial. Watch the video below for further explanation. To practice with polynomials and see how to write them in standard form, watch the

2 following video. Adding and Subtracting Polynomials Polynomials can have operations performed on them just like numbers. We'll start with adding and subtracting. When adding polynomials, you will add like terms together. Remember that like terms have the exact same variable and exponent. If you know how to add polynomials, you will be able to subtract them! In adding polynomials you could add one of two ways...horizontal or vertical. The same is true for subtraction. Also, subtraction of polynomials can be illustrated as adding the opposite. Just like in adding polynomials, you must subtract like terms. Look at some examples in this video. If you need more help with this, see the resources in the sidebar. For more practice, try the Quizlet in the sidebar. Polynomials: Multiply and Divide To multiply polynomials, the distributive property is used; which is...for all real numbers a, b, and c, a(b+c)=ab+ac and (b+c)a=ba+ca. This also is true for subtraction. (Remember, when multiplying like bases, you add the exponents together;.) When you are multiplying 2 binomials, you distribute using a method called "FOIL".

3 Watch these 2 videos to review the steps for multiplying polynomials. ** SPECIAL NOTE: The last problem on the first video below gives an incorrect answer choice. Please change to. If you need more review and practice, go to the sidebar and check out the resources. Binomial Theorem There are times when we have multiplied a binomial by a binomial. We've referred to this process as FOILing or distributing twice. What if we wanted to multiply a binomial by itself more than twice? Multiplying could take a long time. Finding would be very tedious and leave a lot of room for error. Instead, we can use the Binomial Theorem. The formula for the Binomial Theorem is shown here. The binomial theorem states that when raising a binomial to an integer power, the binomial coefficients can be calculated using the combination where n is the integer power, and k is the order of the coefficient. This is expressed

4 mathematically with the expression Basically, this says that the powers of the first term start with the power of the binomial and go down. The powers of the second term start with 0 and go up. And we use combinations to find the coefficients, which can be found using Pascal's Triangle. Watch this video to learn more. When raising a binomial to a power, the degree of each term will be the same as the power of the binomial. For example, in the problem, if you add the powers in each term to get the degree, they all come out to 6, the power of the original binomial. For more review (and a shortcut), watch the Binomial Theorem video in the sidebar. For more on the Binomial Theorem and Pascal's Triangle, go to the sidebar. Dividing Polynomials Let's start with dividing a polynomial by a monomial. The following video will show you the process. When dividing polynomials by something that isn't a monomial, we will use long division or synthetic division. Long division with polynomials is just like the long division that you learned in elementary school but now we will also be using variables. Divide the first term of the dividend by the first term of the divisor, then multiply and subtract. You will have a new polynomial to repeat the original process with. If there is a remainder other than zero, write the remainder as a fraction with the remainder as the numerator and the original divisor as the denominator. Watch this video to learn the steps and walk through some examples. Synthetic division is a method that removes the variables during the division process but puts them back at the end to recreate a polynomial expression. Instead of divide and subtract, you multiply and add. (Any remainders other than zero are treated the same way they are in long division, rewritten as a fraction with the remainder as the numerator and the divisor as the denominator.) This video will take you through the process and show you some examples. In all division, you can always check your solutions by multiplying the quotient by the divisor to get the original polynomial.

5 If you need more practice, check out the videos in the sidebar. Now do the Self Check. Unless a method is specified, you can use either, keeping in mind synthetic division works only with divisor in the form It is possible to synthetic divide if the coefficient of the x is a number other than 1. It does involve dividing with fractions, so usually long division is easier. For more on this, see "Extra" in the sidebar. Operations with Polynomials Assignment Select the "Operations with Polynomials" Handout from the sidebar. Record your answers in a separate document. Submit your completed assignment. Pascal's Triangle Discussion It is now time to complete the " Pascal's Triangle " discussion. A rubric for your discussion in located in the sidebar. In Lesson Topic 2, you learned about Pascal's Triangle. The video completes Pascal's Triangle out to the 5th row. In your post, give the next row of the triangle. Next, research Pascal's Triangle. Find other ways that Pascal's triangle is used, or interesting patterns or facts about the Triangle.

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