Worksheet - Polynomials (Word - ) - Free Printable

Problem Analysis:

The worksheet provided involves identifying whether expressions are polynomials, classifying them as monomials, binomials, or trinomials, determining the number of terms in each polynomial, and finding the degree of the first term of each polynomial. Let's solve each section step by step.

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Section 1: Is each of these a polynomial?

A polynomial is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. Expressions with negative exponents, fractional exponents, or other non-polynomial operations (like square roots of variables) are not polynomials.

#### Solutions:
1. \( 4x^2 - 8x + 13 \)
- This is a polynomial because it consists of terms with non-negative integer exponents.
- Answer: YES

2. \( 3x^5y^4 + 2n^3p^8 - 6r^2 \)
- This is a polynomial because all terms have non-negative integer exponents.
- Answer: YES

3. \( 2x^5 + 3\sqrt{x} - 9 \)
- The term \( 3\sqrt{x} \) is equivalent to \( 3x^{1/2} \), which has a fractional exponent. Therefore, this is not a polynomial.
- Answer: NO

4. \( \frac{5}{x^2} + \frac{2}{y^2} \)
- The terms \( \frac{5}{x^2} \) and \( \frac{2}{y^2} \) have negative exponents (\( x^{-2} \) and \( y^{-2} \)). Therefore, this is not a polynomial.
- Answer: NO

5. \( 7a^5 + 2b^4 - 3c^6 \)
- This is a polynomial because all terms have non-negative integer exponents.
- Answer: YES

6. \( 7^x + 5^{2x+1} - 3 \)
- The terms \( 7^x \) and \( 5^{2x+1} \) involve variables in the exponent, which is not allowed in polynomials. Therefore, this is not a polynomial.
- Answer: NO

7. \( 8g^4k^{13} - 7t^5e^{11} \)
- This is a polynomial because all terms have non-negative integer exponents.
- Answer: YES

8. \( \frac{1}{2}n^3 + \frac{2}{3}n^2 - \frac{5}{7}n - 14 \)
- This is a polynomial because all terms have non-negative integer exponents.
- Answer: YES

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Section 2: Is each of these a monomial, a binomial, or a trinomial?

- A monomial is a single term.
- A binomial is a polynomial with exactly two terms.
- A trinomial is a polynomial with exactly three terms.

#### Solutions:
9. \( 7n^5r^3 + 8n^8 \)
- This has two terms: \( 7n^5r^3 \) and \( 8n^8 \).
- Answer: Binomial

10. \( x^2 + 3x + 2 \)
- This has three terms: \( x^2 \), \( 3x \), and \( 2 \).
- Answer: Trinomial

11. \( 2x^2y^{11}z \)
- This has one term: \( 2x^2y^{11}z \).
- Answer: Monomial

12. \( 3n^5 \)
- This has one term: \( 3n^5 \).
- Answer: Monomial

13. \( 8e + 2 \)
- This has two terms: \( 8e \) and \( 2 \).
- Answer: Binomial

14. \( 5x^4 + 7x^{13} + 11x^7 \)
- This has three terms: \( 5x^4 \), \( 7x^{13} \), and \( 11x^7 \).
- Answer: Trinomial

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Section 3: How many terms does each polynomial have?

Count the number of terms in each polynomial.

#### Solutions:
15. \( 2y^4 + 7y^3 - 8y^2 + 3y - 11 \)
- Terms: \( 2y^4 \), \( 7y^3 \), \( -8y^2 \), \( 3y \), \( -11 \).
- Number of terms: 5.
- Answer: 5

16. \( 4x^2y^{13} + 9x^5y^{11}z \)
- Terms: \( 4x^2y^{13} \), \( 9x^5y^{11}z \).
- Number of terms: 2.
- Answer: 2

17. \( x^{14} + 2x^{13} - 5x^{12} + 3x^{11} - 2x^{10} + 5x^9 + 7x^8 - x^7 + 3x^6 + 11x^5 + 2x^4 - 7x^3 - 5x^2 + 8x + 9 \)
- Count all terms: There are 15 terms.
- Answer: 15

18. \( 12c^2d + 3d^5x - 2n^4h + 3 \)
- Terms: \( 12c^2d \), \( 3d^5x \), \( -2n^4h \), \( 3 \).
- Number of terms: 4.
- Answer: 4

19. \( 4 + 5e^2g - 2f^2 + 6x - 13k^5 \)
- Terms: \( 4 \), \( 5e^2g \), \( -2f^2 \), \( 6x \), \( -13k^5 \).
- Number of terms: 5.
- Answer: 5

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Section 4: What is the degree of the first term of each polynomial?

The degree of a term is the sum of the exponents of the variables in that term. Identify the first term and calculate its degree.

#### Solutions:
20. \( 2y^4 + 7y^3 - 8y^2 + 3y - 11 \)
- First term: \( 2y^4 \).
- Degree: 4.
- Answer: 4

21. \( 4x^2y^{13}z + 9x^5y^{11}z \)
- First term: \( 4x^2y^{13}z \).
- Degree: \( 2 + 13 + 1 = 16 \).
- Answer: 16

22. \( x^{14} + 2x^{13} - 5x^{12} + 3x^{11} - 2x^{10} + 5x^9 + 7x^8 - x^7 + 3x^6 + 11x^5 + 2x^4 - 7x^3 - 5x^2 + 8x + 9 \)
- First term: \( x^{14} \).
- Degree: 14.
- Answer: 14

23. \( 12c^2d + 3d^5x - 2n^4h + 3 \)
- First term: \( 12c^2d \).
- Degree: \( 2 + 1 = 3 \).
- Answer: 3

24. \( 4 + 5e^2g - 2f^2 + 6x - 13k^5 \)
- First term: \( 4 \) (constant term).
- Degree: 0.
- Answer: 0

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Final Answers:

1. YES
2. YES
3. NO
4. NO
5. YES
6. NO
7. YES
8. YES

9. Binomial
10. Trinomial
11. Monomial
12. Monomial
13. Binomial
14. Trinomial

15. 5
16. 2
17. 15
18. 4
19. 5

20. 4
21. 16
22. 14
23. 3
24. 0

Boxed Final Answer:
\[
\boxed{
\begin{array}{l}
\text{YES, YES, NO, NO, YES, NO, YES, YES} \\
\text{Binomial, Trinomial, Monomial, Monomial, Binomial, Trinomial} \\
\text{5, 2, 15, 4, 5} \\
\text{4, 16, 14, 3, 0}
\end{array}
}
\]