1 Topic 6.1.1 Polynomials. 2 Lesson 1.1.1 California Standard: 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve

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Topic 6.1.1Topic 6.1.1

PolynomialsPolynomials

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Lesson

1.1.1

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.

PolynomialsPolynomialsTopic

6.1.1

Key words:• monomial• polynomial• like terms• degree

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Lesson

1.1.1

Polynomial — another math word that sounds a lot harder than it actually is.


6.1.1

Read on and you’ll see that actually polynomials are not as complicated as you might think.

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Lesson

1.1.1

A Monomial is a Single Term


6.1.1

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.

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Lesson

1.1.1

A Polynomial Can Have More Than One Term


6.1.1

A polynomial is an algebraic expression that has one or more terms (each of which is a monomial).

There are a couple of special types of polynomial:

For example, x + 1 and –3x2 + 2x + 1 are polynomials.

A binomial is a two-term polynomial, such as x2 + 1.

A trinomial is a polynomial with three terms, such as –3x2 + 2x + 1.

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For each of the polynomials below, state whether it is a monomial, a binomial, or a trinomial.

Lesson

1.1.1

Guided Practice


6.1.1

Solution follows…

1. 5x 2. 9x2 + 4

3. 2y 4. 2x2y

5. 2y + 2 6. 5x – 2

7. 7x3 8. x2 + y

monomial

binomial

monomialmonomial

monomial

binomial

binomial

binomial

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For each of the polynomials below, state whether it is a monomial, a binomial, or a trinomial.

Lesson

1.1.1

Guided Practice


6.1.1

Solution follows…

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 binomial

monomial

trinomial

trinomial

monomial

binomial

trinomial

trinomial

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Lesson

1.1.1

Use Like Terms to Simplify Polynomials


6.1.1

Like terms are terms that have exactly the same variables — for example, –2x2 and 5x2 are like terms.

A polynomial can often be simplified by combining all like terms.

Like terms always have the same variables, but may have different coefficients.

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Example 1

Topic

6.1.1

Simplify the expression 2x2 + 4y + 3x2.

Solution

Notice that there are two like terms, 2x2 and 3x2:

Solution follows…

You can combine the like terms: 2x2 + 3x2 = 5x2

So 2x2 + 4y + 3x2 = 5x2 + 4y

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Simplify each of the following polynomials.

Lesson

1.1.1

Guided Practice


6.1.1

Solution follows…

17. x + 1 + 2x 18. 3y + 2y

19. 9x2 + 4x + 7x 20. x2 + x + x2 2x2 + x

5y3x + 1

9x2 + 11x

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Simplify each of the following polynomials, then state whether your answer is a monomial, binomial, or trinomial.

Lesson

1.1.1

Guided Practice


6.1.1

Solution follows…

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

x2y + 8, binomial

x4 + 2x3 + 4, trinomial

–2y, monomial

0, monomial

14x2 + 2y + 4, trinomial

7xy + 5x2y – 4xy2, trinomial

14x5 + x4 + x2, trinomial

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Lesson

1.1.1

Finding the Degree of a Polynomial


6.1.1

The degree of a polynomial in x is the size of the highest power of x in the expression.

For example:

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.

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

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State the degree of each of the following polynomials.

Lesson

1.1.1

Guided Practice


6.1.1

Solution follows…

29. 3x + 5 30. 3x4 + 2

31. x2 + 2x + 3 32. x + 4 + 2x22nd degree 2nd degree

1st degree 4th degree

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Simplify and state the degree of each of the following polynomials.

Lesson

1.1.1

Guided Practice


6.1.1

Solution follows…

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

–2x9 – 8x4 + 13x2 + x, 9th degreec, 1st degree

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For the polynomials below state whether they are a monomial, a binomial, or a trinomial.


Independent Practice

Solution follows…

Topic

6.1.1

1. 19a2 + 16 2. 2c – 4a + 6

3. 42xy 4. 16a2b + 4ab2

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

binomial

binomialmonomial

trinomial

–0.3x2 + 9.8

10x2

–8.2x4 + 0.9x2 + 9.6

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State the degree of the following polynomials.



Solution follows…

Topic

6.1.1

8. x – 9x6 + 4 9. 14x8 + 16x10 + 4x8

10. 2x2 – 4x4 + 7x5 11. 2x2 – 4x + 8

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

6th degree

2nd degree5th degree

10th degree

12a2 – 4a + 176, 2nd degree, trinomial

–14.5x4 + 9.3x2, 4th degree, binomial

0, degree 0, monomial

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Solution follows…

Topic

6.1.1

17. When a third degree monomial is added to a second degree binomial, what is the result?

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 –

79

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12

13

34

35

34

19

x9 – 2x3 + , 9th degree, trinomial79

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18. When a 4th degree monomial is added to a 6th degree binomial, what are the possible results?

A third degree trinomial

6th degree polynomial that could be either trinomial, binomial, or monomial

29x10 – x6 – , 10th degree,

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2720

trinomial

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Topic

6.1.1

Round UpRound 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.

Documents

1 Topic 6.1.1 Polynomials. 2 Lesson 1.1.1 California Standard: 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve