Chapter 2 Algebraic Expression. Adding and Subtracting Polynomials
- 2. • Adding and Subtracting Polynomials • Multiplying Algebraic Expressions • Special Product Formulas • Factoring Common Factors • Factoring Trinomials • Special Factoring Formulas
- 3. A variable is a letter that can represent any number from a given set of numbers. If we start with variables, such as x, y, and z, and some real numbers and combine them using addition, subtraction, multiplication, division, powers, and roots, we
- 4. A monomial is an expression of the form axk , where a is a real number and k is a nonnegative integer. A binomial is a sum of two monomials and a trinomial is a sum of three monomials. In general, a sum of monomials is
- 7. 1. Adding and Subtracting Polynomials We add and subtract polynomials using the properties of real numbers that were discussed in Chapter 1. The idea is to combine like terms (that is, terms with the same variables raised to the same powers) using the Distributive
- 8. 1. Adding and Subtracting Polynomials In subtracting polynomials, we have to remember that if a minus sign precedes an expression in parentheses, then the sign of every term within the parentheses is changed when we remove the parentheses.
- 9. Find the sum: (𝑥3 −6 𝑥2 +2 𝑥 +4)+( 𝑥3 +5 𝑥2 −7 𝑥 ) Find the difference: (𝑥 3 −6 𝑥 2 +2 𝑥 +4)− (𝑥 3 +5 𝑥 2 − 7 𝑥)
- 10. 2. Multiplying Algebraic Expressions To find the product of polynomials or other algebraic expressions, we need to use the Distributive Property repeatedly. In particular, using it three times on the product of two binomials, we get:
- 11. 2. Multiplying Algebraic Expressions This says that we multiply the two factors by multiplying each term in one factor by each term in the other factor and adding these products. Schematically, we have
- 12. Find the product: (2 𝑥+3 ) ( 𝑥 2 −5 𝑥+ 4 )
- 13. 2. Multiplying Algebraic Expressions When we multiply trinomials or other polynomials with more terms, we use the Distributive Property. It is also helpful to arrange our work in table form.
- 14. 2. Multiplying Algebraic Expressions In general, we can multiply two algebraic expressions by using the Distributive Property and the Laws of Exponents.
- 15. 3. Special Product Formulas
- 16. Find the product: (2 𝑥 − √ 𝑦 )(2 𝑥+√ 𝑦 ) ( 𝑥+ 𝑦 −1) (𝑥 + 𝑦 +1) ( 3 𝑥 + 5 )2 ( 𝑥 2 +5 ) 3
- 17. 4. Factoring Common Factors we use the Distributive Property to expand algebraic expressions. We sometimes need to reverse this process (again using the Distributive Property) by factoring an expression as a
- 18. 4. Factoring Common Factors The easiest type of factoring occurs when the terms have a common factor.
- 19. Factor each expression: 3 𝑥2 − 6𝑥 8 𝑥4 𝑦2 +6 𝑥3 𝑦3 − 2𝑥𝑦4 (2 𝑥+4 )( 𝑥 − 3) −5 ( 𝑥 −3 )
- 20. 5. Factoring Trinomials To factor a trinomial of the form 𝑥2 +𝑏𝑥+𝑐 We note that ( 𝑥+ 𝑟 ) (𝑥 + 𝑠)=𝑥2 +(𝑟 + 𝑠 ) 𝑥 +𝑟𝑠 So we need to choose numbers r and s so that r + s = b and rs = c
- 21. Factor: 6 𝑥2 +7 𝑥 −5 𝑥2 −2 𝑥 − 3 (5 𝑎+1)2 −2 (5 𝑎+ 1)− 3
- 22. 6. SPECIAL Factoring FORMULAs
- 23. Factor each expression: 4 𝑥2 −25 ( 𝑥+ 𝑦 )2 − 𝑧 2
- 25. Factor each trinomials: 𝑥2 +6 𝑥+9 4 𝑥2 − 4 𝑥𝑦+𝑦2
- 26. Factor each polynomials: 27 𝑥3 − 1 𝑥 6 +8
- 27. Factor each expression: 2 𝑥4 − 8 𝑥2 𝑥5 𝑦 2 − 𝑥 𝑦6
- 28. 7. Factoring by Grouping Terms Polynomials with at least four terms can sometimes be factored by grouping terms. The following example illustrates the idea.
- 29. Factor each polynomial 𝑥3 +𝑥2 +4 𝑥+4 𝑥3 − 2𝑥2 − 9𝑥+18