Polynomials Topic 6.1.1.

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Presentation transcript:

Polynomials Topic 6.1.1

Polynomials 6.1.1 1.1.1 California Standard: What it means for you: Lesson 1.1.1 Topic 6.1.1 Polynomials California Standard: 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques. What it means for you: You’ll learn what polynomials are, and you’ll simplify them. Key words: monomial polynomial like terms degree

Topic 6.1.1 Lesson 1.1.1 Polynomials Polynomial — another math word that sounds a lot harder than it actually is. Read on and you’ll see that actually polynomials are not as complicated as you might think.

Polynomials 6.1.1 1.1.1 A Monomial is a Single Term Lesson 1.1.1 Topic 6.1.1 Polynomials A Monomial is a Single Term A monomial is a single-term expression. It can be either a number or a product of a number and one or more variables. For example, 13, 2x2, and –x3yn4 are all monomials.

A binomial is a two-term polynomial, such as x2 + 1. Topic 6.1.1 Lesson 1.1.1 Polynomials A Polynomial Can Have More Than One Term A polynomial is an algebraic expression that has one or more terms (each of which is a monomial). For example, x + 1 and –3x2 + 2x + 1 are polynomials. There are a couple of special types of polynomial: A binomial is a two-term polynomial, such as x2 + 1. A trinomial is a polynomial with three terms, such as –3x2 + 2x + 1.

6.1.1 1.1.1 Polynomials Guided Practice Topic 6.1.1 Lesson 1.1.1 Polynomials Guided Practice For each of the polynomials below, state whether it is a monomial, a binomial, or a trinomial. 1. 5x 2. 9x2 + 4 3. 2y 4. 2x2y 5. 2y + 2 6. 5x – 2 7. 7x3 8. x2 + y monomial binomial monomial monomial binomial binomial monomial binomial Solution follows…

Polynomials 6.1.1 1.1.1 Guided Practice Lesson 1.1.1 Topic 6.1.1 Polynomials Guided Practice For each of the polynomials below, state whether it is a monomial, a binomial, or a trinomial. 9. x2 + 2x + 3 10. 3x2 + 4x – 8 11. 4.3x – 8.9x2 + 4.2x3 12. 0.3x2y 13. 0.3x2y + xy2 + 4xy 14. 8.7 15. a2 + b2 16. 97.9a – 14.2c trinomial trinomial trinomial monomial trinomial monomial binomial binomial Solution follows…

Polynomials 6.1.1 1.1.1 Use Like Terms to Simplify Polynomials Lesson 1.1.1 Topic 6.1.1 Polynomials Use Like Terms to Simplify Polynomials Like terms are terms that have exactly the same variables — for example, –2x2 and 5x2 are like terms. Like terms always have the same variables, but may have different coefficients. A polynomial can often be simplified by combining all like terms.

Polynomials 6.1.1 Simplify the expression 2x2 + 4y + 3x2. Solution Topic 6.1.1 Polynomials Example 1 Simplify the expression 2x2 + 4y + 3x2. Solution Notice that there are two like terms, 2x2 and 3x2: You can combine the like terms: 2x2 + 3x2 = 5x2 So 2x2 + 4y + 3x2 = 5x2 + 4y Solution follows…

Polynomials 6.1.1 1.1.1 Guided Practice Lesson 1.1.1 Topic 6.1.1 Polynomials Guided Practice Simplify each of the following polynomials. 17. x + 1 + 2x 18. 3y + 2y 19. 9x2 + 4x + 7x 20. x2 + x + x2 3x + 1 5y 9x2 + 11x 2x2 + x Solution follows…

6.1.1 1.1.1 Polynomials Guided Practice Topic 6.1.1 Lesson 1.1.1 Polynomials Guided Practice Simplify each of the following polynomials, then state whether your answer is a monomial, binomial, or trinomial. 21. 3x2 + 4 – 8 + x2 22. 8x3 + x4 – 6x3 + 4 23. 3x2y – 2x2y + 8 24. 7 – 2y + 3 – 10 25. 4x3 + 7 – x3 + 4 – 3x3 – 11 26. 5x2 + 9x2 + 4 + 2y 27. 3xy + 4xy + 5x2y – 4xy2 28. 9x5 + 2x2 + 4x4 + 5x5 – 3x4 – x2 4x2 – 4, binomial x4 + 2x3 + 4, trinomial x2y + 8, binomial –2y, monomial 0, monomial 14x2 + 2y + 4, trinomial 7xy + 5x2y – 4xy2, trinomial 14x5 + x4 + x2, trinomial Solution follows…

6.1.1 1.1.1 Polynomials Finding the Degree of a Polynomial Topic 6.1.1 Lesson 1.1.1 Polynomials Finding the Degree of a Polynomial The degree of a polynomial in x is the size of the highest power of x in the expression. If you see the phrase “a fourth-degree polynomial in x,” you know that it will contain at least one term with x4, but it won’t contain any higher powers of x than 4. For example: 2x + 1 has degree 1 — it’s a first-degree polynomial in x y2 + y – 3 has degree 2 — it’s a second-degree polynomial in y x4 – x2 has degree 4 — it’s a fourth-degree polynomial in x

Polynomials 6.1.1 1.1.1 Guided Practice Lesson 1.1.1 Topic 6.1.1 Polynomials Guided Practice State the degree of each of the following polynomials. 29. 3x + 5 30. 3x4 + 2 31. x2 + 2x + 3 32. x + 4 + 2x2 1st degree 4th degree 2nd degree 2nd degree Solution follows…

6.1.1 1.1.1 Polynomials Guided Practice Topic 6.1.1 Lesson 1.1.1 Polynomials Guided Practice Simplify and state the degree of each of the following polynomials. 33. 2x + x2 + x – 3 34. 3a3 + 4a – 2a3 + 4a2 35. 4x3 + 4x8 – 3x8 + 2x3 36. 3y + 2y – 5y2 + 6y 37. b13 + 2b13 – 8 + 4 – 3b13 38. z3 + z3 – z6 + z7 + 3z7 39. c4 + c3 + c3 – c4 + c – 2c3 40. x – 2x9 – 8x4 + 13x2 x2 + 3x – 3, 2nd degree a3 + 4a2 + 4a, 3rd degree x8 + 6x3, 8th degree –5y2 + 11y, 2nd degree –4, degree 0 4z7 – z6 + 2z3, 7th degree c, 1st degree –2x9 – 8x4 + 13x2 + x, 9th degree Solution follows…

Polynomials 6.1.1 Independent Practice Topic 6.1.1 Polynomials Independent Practice For the polynomials below state whether they are a monomial, a binomial, or a trinomial. 1. 19a2 + 16 2. 2c – 4a + 6 3. 42xy 4. 16a2b + 4ab2 binomial trinomial monomial binomial Simplify each of the following polynomials. 5. 0.7x2 + 9.8 – x2 6. 17x2 – 14x9 + 7x9 – 7x2 + 7x9 7. 0.8x4 + 0.3x2 + 9.6 – x2 – 9x4 + 1.6x2 –0.3x2 + 9.8 10x2 –8.2x4 + 0.9x2 + 9.6 Solution follows…

Polynomials 6.1.1 Independent Practice Topic 6.1.1 Polynomials Independent Practice State the degree of the following polynomials. 8. x – 9x6 + 4 9. 14x8 + 16x10 + 4x8 10. 2x2 – 4x4 + 7x5 11. 2x2 – 4x + 8 6th degree 10th degree 5th degree 2nd degree Simplify each polynomial, state the degree of the polynomial, and determine whether it is a monomial, a binomial, or a trinomial. 12. 93a2 + 169 – 4a – 81a2 + 7 13. 7.9x2 – 13x4 – 1.5x4 + 1.4x2 14. 5x9 – 6x9 + 4 + x9 – 3 – 1 12a2 – 4a + 176, 2nd degree, trinomial –14.5x4 + 9.3x2, 4th degree, binomial 0, degree 0, monomial Solution follows…

x9 – 2x3 + , 9th degree, trinomial Topic 6.1.1 Polynomials Independent Practice Simplify each polynomial, state the degree of the polynomial, and determine whether it is a monomial, a binomial, or a trinomial. 15. x9 – x3 – x3 + x9 + 16. x10 – x6 – x6 + x10 – 7 9 1 2 3 4 5 x9 – 2x3 + , 9th degree, trinomial 7 9 5 6 2 9 x10 – x6 – , 10th degree, 3 4 27 20 trinomial 17. When a third degree monomial is added to a second degree binomial, what is the result? A third degree trinomial 18. When a 4th degree monomial is added to a 6th degree binomial, what are the possible results? 6th degree polynomial that could be either trinomial, binomial, or monomial Solution follows…

Topic 6.1.1 Polynomials Round Up This Topic gets you started on manipulating polynomials, by simplifying them. In the next couple of Topics you’ll see how to add and subtract polynomials.