Review on Polynomial worksheet - Free Printable

Let's solve the problem step by step.

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🔷 What is a Polynomial?

A polynomial is an algebraic expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

#### ✔ Rules for a polynomial:
- Only non-negative integer exponents (like 0, 1, 2, 3, ...)
- No variables in denominators, no roots, no negative exponents, no fractional exponents

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We'll go through each expression one by one.

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1. $ 2a^3 + 3a^4 + 18a + a^5 - 9 $

- All exponents are non-negative integers → ✔ Polynomial
- Rewrite in standard form: $ a^5 + 3a^4 + 2a^3 + 18a - 9 $
- Degree: Highest exponent = 5
- Leading coefficient: Coefficient of $ a^5 $ = 1
- Number of terms: 5

✔ Polynomial

| Given | Degree | Leading Coefficient | Number of Terms |
|-------|--------|---------------------|------------------|
| $ 2a^3 + 3a^4 + 18a + a^5 - 9 $ | 5 | 1 | 5 |

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2. $ \sqrt{3a^2} - 3a + 1 $

- Simplify $ \sqrt{3a^2} = \sqrt{3} \cdot a $ → This is $ \sqrt{3}a $
- But $ \sqrt{3a^2} $ is equivalent to $ \sqrt{3} \cdot |a| $, which is not a polynomial because it involves a square root of a variable.
- Even though it simplifies to linear in $ a $, the original form has a radical, so it's not considered a polynomial unless simplified properly.

⚠️ Not a polynomial due to the square root.

| Given | Degree | Leading Coefficient | Number of Terms |
|-------|--------|---------------------|------------------|
| $ \sqrt{3a^2} - 3a + 1 $ | ✘ Not a polynomial | — | — |

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3. $ 3a^3 - a^{-2} + 2a - 1 $

- Contains $ a^{-2} $ → negative exponent
- ✘ Not allowed in polynomials

✘ Not a polynomial

| Given | Degree | Leading Coefficient | Number of Terms |
|-------|--------|---------------------|------------------|
| $ 3a^3 - a^{-2} + 2a - 1 $ | ✘ Not a polynomial | — | — |

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4. $ \frac{1}{a^2} - 3a^3 + 4a - 3 $

- $ \frac{1}{a^2} = a^{-2} $ → negative exponent
- ✘ Not a polynomial

✘ Not a polynomial

| Given | Degree | Leading Coefficient | Number of Terms |
|-------|--------|---------------------|------------------|
| $ \frac{1}{a^2} - 3a^3 + 4a - 3 $ | ✘ Not a polynomial | — | — |

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5. $ x^2 - 9 $

- Only positive integer exponents → ✔ Polynomial
- Degree: 2
- Leading coefficient: 1 (coefficient of $ x^2 $)
- Number of terms: 2

✔ Polynomial

| Given | Degree | Leading Coefficient | Number of Terms |
|-------|--------|---------------------|------------------|
| $ x^2 - 9 $ | 2 | 1 | 2 |

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6. $ 11x + 10x^2 + 3 $

- All exponents are non-negative integers → ✔ Polynomial
- Standard form: $ 10x^2 + 11x + 3 $
- Degree: 2
- Leading coefficient: 10
- Number of terms: 3

✔ Polynomial

| Given | Degree | Leading Coefficient | Number of Terms |
|-------|--------|---------------------|------------------|
| $ 11x + 10x^2 + 3 $ | 2 | 10 | 3 |

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7. $ y^4 - 2y^3 - 13y^2 + 16y + 24 $

- All exponents are non-negative integers → ✔ Polynomial
- Degree: 4
- Leading coefficient: 1 (coefficient of $ y^4 $)
- Number of terms: 5

✔ Polynomial

| Given | Degree | Leading Coefficient | Number of Terms |
|-------|--------|---------------------|------------------|
| $ y^4 - 2y^3 - 13y^2 + 16y + 24 $ | 4 | 1 | 5 |

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8. $ 30 + y^3 + y - 6y^2 $

- Rearranged: $ y^3 - 6y^2 + y + 30 $
- All exponents are non-negative integers → ✔ Polynomial
- Degree: 3
- Leading coefficient: 1
- Number of terms: 4

✔ Polynomial

| Given | Degree | Leading Coefficient | Number of Terms |
|-------|--------|---------------------|------------------|
| $ 30 + y^3 + y - 6y^2 $ | 3 | 1 | 4 |

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9. $ 17a - 11a^2 - 25 + 2a^3 $

- Rearranged: $ 2a^3 - 11a^2 + 17a - 25 $
- All exponents are non-negative integers → ✔ Polynomial
- Degree: 3
- Leading coefficient: 2
- Number of terms: 4

✔ Polynomial

| Given | Degree | Leading Coefficient | Number of Terms |
|-------|--------|---------------------|------------------|
| $ 17a - 11a^2 - 25 + 2a^3 $ | 3 | 2 | 4 |

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10. $ -3 $

- Constant term → this is a constant polynomial
- Degree: 0 (since it's a constant)
- Leading coefficient: -3
- Number of terms: 1

✔ Polynomial

| Given | Degree | Leading Coefficient | Number of Terms |
|-------|--------|---------------------|------------------|
| $ -3 $ | 0 | -3 | 1 |

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✔ Final Answer Table:

| GIVEN | DEGREE | LEADING COEFFICIENT | NUMBER OF TERMS |
|------|--------|---------------------|------------------|
| $ 2a^3 + 3a^4 + 18a + a^5 - 9 $ | 5 | 1 | 5 |
| $ \sqrt{3a^2} - 3a + 1 $ | ✘ Not a polynomial | — | — |
| $ 3a^3 - a^{-2} + 2a - 1 $ | ✘ Not a polynomial | — | — |
| $ \frac{1}{a^2} - 3a^3 + 4a - 3 $ | ✘ Not a polynomial | — | — |
| $ x^2 - 9 $ | 2 | 1 | 2 |
| $ 11x + 10x^2 + 3 $ | 2 | 10 | 3 |
| $ y^4 - 2y^3 - 13y^2 + 16y + 24 $ | 4 | 1 | 5 |
| $ 30 + y^3 + y - 6y^2 $ | 3 | 1 | 4 |
| $ 17a - 11a^2 - 25 + 2a^3 $ | 3 | 2 | 4 |
| $ -3 $ | 0 | -3 | 1 |

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📝 Summary:

- A polynomial must have only non-negative integer exponents.
- Expressions with negative exponents, fractional exponents, or roots of variables are not polynomials.
- Constants are polynomials of degree 0.
- The leading coefficient is the coefficient of the highest-degree term.
- The number of terms is simply how many separate parts there are (after combining like terms).

Let me know if you'd like this as a downloadable table!