Pre-Algebra Notes Unit 1: Polynomials and Sequences Polynomials Syllabus Objective: (6.1) The student will write polynomials in standard form. Let s review a definition: monomial. A monomial is a number, a variable, or a product of numbers and variables where eponents are whole numbers. Take a moment to look at samples of epressions that are and are not monomials. monomials 4 6 4 3p q 3 3. NOT monomials y a A polynomial is a monomial or a sum of monomials. We will define each monomial in a polynomial as a term. Polynomials can be classified by their number of terms: monomial has 1 term, binomial has terms and a trinomial has 3 terms. Eample: Classify each epression. a) 5 3 5y binomial; polynomial with terms b) 3 4a b monomial; polynomial with 1 term c) 1 6 6 not a polynomial; eponent is not a whole number d) y 4b 7 trinomial; polynomial with 3 terms The degree of a term is the sum of the eponents of its variables. We will define the degree of the polynomial as the greatest degree of its terms. Note that the degree of a constant is 0. Eample: Find the degree of each polynomial. a) 5 9 degree ; 5 has degree, has degree 1, 9 has degree of 0 the greatest degree is. b) 3 3 y degree 5; 3 has degree 3, y 3 has degree + 3 = 5, has degree 1 the greatest degree is 5. Polynomials should be written in standard form. A polynomial is written in standard form if it is simplified (all like terms are combined) and the terms are arranged in descending order by degree. McDougal Littell, Chapter 1, Sections 1-3, & 8 Pre-Algebra Unit 1: Polynomials and Sequences Page 1 of 7
Eample: Write 4 6 4 3 as a polynomial in standard form. 3 4 6 4 3 = 3 4 1 4 3 distribute 3 4 3 1 4 group like terms 5 7 16 combine like terms 5 16 7 put into standard form Adding and Subtracting Polynomials Syllabus Objectives: (6.) The student will add polynomials. (6.3) The students will subtract polynomials. Polynomials can sound tough. But if you look at them closely, you will notice that students have worked with polynomial epressions such as 6 5 since first grade. The only difference is that they have letters ( s) instead of powers of ten. They have been taught that 65 means 6(100) + 5(10) + (1). They have been taught the si tells them how many hundreds they have, the five how many tens, and the two how many ones are in the number. 6(100) + 5(10) + (1) 610 510 1 6 5 The point is this: polynomial epressions in algebra are linked to what is referred to as epanded notation in grade school. It s not a new concept. In grade school we teach the students how to add or subtract numbers using place value. Typically, we have them line up the numbers vertically so the ones digits are in a column, the tens digits are in the net column and so on; then we have them add or subtract from right to left. In algebra, we have the students line up the polynomials the same way; we line up the numbers, the s, and the s, then perform the operation as shown below. 443 5 + 35 5 7 8 578 Notice when adding, we added like terms. That is, with the numbers, we added the hundreds column to the hundreds, the tens to the tens, the ones to ones. In algebra, we add the s to the s, the s to the s, the constants to the constants. In algebra, the students can add the epressions from right to left as they have been taught or left to right. If the students understand place value, this could lead students to add columns of numbers more quickly, without regrouping, by adding numbers from left to right. McDougal Littell, Chapter 1, Sections 1-3, & 8 Pre-Algebra Unit 1: Polynomials and Sequences Page of 7
The students would have to add the hundreds column, then add to that the sum of the tens column, and finally the sum of the ones column. Eample: Add, in your head 341 + 14 + 13. You have 300 + 00 + 100, that s 600. Adding the tens, we have 40 + 10 + 30 which now gives us 680. Finally adding 1 + 4 + or 7, the sum is 687. We can add, subtract, multiply, and divide polynomials using the same procedures we learned in elementary school. In the first grade you learned to add the ones column to the ones column, the tens to the tens, hundreds to hundreds. We use that same concept to add polynomials: we add numbers to numbers, s to s, and s to s. Of course, this is combining like terms. Eample: Add 4 5 7 6. There are ways to look at this; adding horizontally or vertically. Horizontal method: 4 5 76 5 74 6 Group like terms 4 5 7 6 Associative Property 8 5 10 Combine like terms Vertical method: 4 5 76 Arrange terms in columns of like terms 8 5 10 Combine like terms Subtraction can be tricky remember to change ALL the signs following a subtraction sign, then add. Eample: Subtract 3 4 1 5 3 6. Again, there are ways to look at this; subtracting horizontally or vertically. McDougal Littell, Chapter 1, Sections 1-3, & 8 Pre-Algebra Unit 1: Polynomials and Sequences Page 3 of 7
Horizontal method: 41 5 6 4 1 5 6 Add the opposite 415 6 Associative Property 5 4 1 6 Group like terms Vertical method: 41 5 6 3 4 7 Combine like terms 4 1 4 1 5 6 5 6 Add the opposite 3 4 7 Combine like terms All too often students do not realize that a rule or procedure they are learning is nothing more than a shortcut. For that reason, math is like magic for many students. If we take the time to develop the patterns, students will not be so easily befuddled. Multiplying Polynomials Syllabus Objective: (6.4) The student will multiply a monomial and a polynomial. Polynomials are multiplied the very same way students learned to multiply in third and fourth grades. Unfortunately, they don t realize it. In grade school, students are taught to line up the numbers vertically. In algebra, students are typically taught to multiply horizontally. Let s take a moment to investigate the relationships. Make the connection that multiplication is repeated addition. (3) 3 ( 3) 3 +3 3 64 6 Use the multiplication algorithm to help students learn polynomial multiplication by stacking the factors. 7(1) ( 5) 1 5 7 147 6 10 McDougal Littell, Chapter 1, Sections 1-3, & 8 Pre-Algebra Unit 1: Polynomials and Sequences Page 4 of 7
Another eample: 131(4) 131 4 584 3 y(4 y3z1) 4 y3z1 3y 1 6 9 3 y y yz y Of course we can multiply horizontally by applying the distributive property, but it may help some students to see the connections to processes with which they are already comfortable. Let s also take a moment to review a rule of eponents: Product of Powers Property: a a a m n mn 4 4 6 3 4 34 7 Eample: Find the product. 4 3 3 Horizontal method: or Vertical method: 4 3 3 4 3 33 3 1 9 5 3 4 4 3 3 3 3 1 9 5 3 Eample: Simplify yy 4y. Horizontal method: or Vertical method: y y y y y yy 4y = 4 4y y8y 3 3 3 y y 4 y y4y y 4 y y8 y 3 3 3 McDougal Littell, Chapter 1, Sections 1-3, & 8 Pre-Algebra Unit 1: Polynomials and Sequences Page 5 of 7
According to the CCSD Pre-Algebra benchmarks, we now skip to: Sequences Syllabus Objective: (6.5) The student will etend arithmetic and geometric sequences. A sequence is an ordered list of numbers (called terms). In an arithmetic sequence, the difference between one term and the net is always the same. This difference is called the common difference. We can add the common difference to each term to get the net term. Eample: Find the common difference in the sequence 7, 11, 15, 19, 7, 11, 15, 19 The terms increase by 4, so the common difference is 4. +4 +4 +4 Eample: Find the common difference in the sequence 6, 3, 0, 3, 6, 3, 0, 3, The terms decrease by 3, so the common difference is 3. 3 3 3 Eample: Find the net three terms in the arithmetic sequence.5, 4, 5.5, 7,.5, 4, 5.5, 7, Each term is 1.5 more than the previous, so to find the net 3 +1.5 +1.5 +1.5 terms 7, 8.5, 10, 11.5 +1.5 +1.5 +1.5 The net three terms are 8.5, 10, and 11.5. In a geometric sequence, the ratio of one term to the net is always the same. This ratio is called the common ratio; each term in a geometric sequence is multiplied by the common ratio to get the net term. For the following eamples, determine if the sequence is geometric. If so, give the common ratio. Eample:, 6, 18, 54,, 6, 18, 54 The sequence is geometric with a common ratio of 3. 3 3 3 McDougal Littell, Chapter 1, Sections 1-3, & 8 Pre-Algebra Unit 1: Polynomials and Sequences Page 6 of 7
Eample: 1, 1, 1, 1, 1 1 1 1,,, The sequence is geometric with a common ratio of 1. 1 1 1 Eample: 16, 8, 4,, 16, 8, 4,, The sequence is geometric with a common ratio 1 1 1 1 of. As with the arithmetic sequence, we should know how to etend a geometric sequence. Eample: Find the net three terms for the geometric sequence 16, 54, 18, 6, The common ratio is 1 ; therefore the net 3 terms 3 6,,, 3 9 The net three terms in the sequence would be,, 3 9. 1 3 1 3 1 3 McDougal Littell, Chapter 1, Sections 1-3, & 8 Pre-Algebra Unit 1: Polynomials and Sequences Page 7 of 7