Grade 9 math activity worksheet on identifying polynomials and analyzing their properties.

Of course! Let's solve this step by step. The task is to determine whether each given expression is a polynomial or not. If it is a polynomial, we must find its:

- Degree (the highest exponent of the variable),
- Leading coefficient (the coefficient of the term with the highest degree),
- Number of terms (how many separate parts are added or subtracted).

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📌 What is a Polynomial?

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

✘ NOT polynomials if they have:
- Negative exponents (e.g., `a⁻²`)
- Fractional exponents (e.g., `√a = a^(1/2)`)
- Variables in the denominator (e.g., `1/a²`)

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Let’s go row by row:

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1. `2a³ + 3a⁴ + 18a + a⁵ - 9`

✔ Polynomial

- Rearrange in descending order: `a⁵ + 3a⁴ + 2a³ + 18a - 9`
- Degree: 5 (highest exponent)
- Leading coefficient: 1 (coefficient of `a⁵`)
- Number of terms: 5

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2. `√(3a²) - 3a + 1`

Simplify: `√(3a²) = √3 * |a|`, but even without simplifying, note that `√(a²)` is equivalent to `|a|`, which is not a polynomial function because it involves absolute value (piecewise definition), and more fundamentally, the square root is a fractional exponent: `(3a²)^(1/2) = √3 * a` — wait, actually, if we assume `a ≥ 0`, then `√(3a²) = √3 * a`, which is linear.

But strictly speaking, in algebra, unless specified, expressions like `√(a²)` are considered to involve radicals or absolute values, and radicals with variables inside are not allowed in polynomials. Even though `√(a²)` simplifies to `|a|`, which is not a polynomial (because of the corner at 0), we usually say such expressions are not polynomials.

✔ However, if we interpret `√(3a²)` as `√3 * a` (assuming `a ≥ 0`), then it becomes a linear term. But since the problem doesn’t specify domain restrictions, and the expression contains a radical, we classify it as NOT a polynomial.

> 💡 Important: In most Algebra 1 contexts, any expression with a radical over a variable is not considered a polynomial.

🚫 Not a polynomial

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3. `3a³ - a⁻² + 2a - 1`

✘ Has a negative exponent (`a⁻²`). Not allowed in polynomials.

🚫 Not a polynomial

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4. `1/a² - 3a³ + 4a - 3`

✘ Has `1/a² = a⁻²` → negative exponent → not a polynomial

🚫 Not a polynomial

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5. `x² - 9`

✔ Polynomial

- Terms: `x²`, `-9`
- Degree: 2
- Leading coefficient: 1
- Number of terms: 2

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6. `11x + 10x² + 3`

✔ Polynomial

- Rearrange: `10x² + 11x + 3`
- Degree: 2
- Leading coefficient: 10
- Number of terms: 3

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7. `y⁴ - 2y³ - 13y² + 16y + 24`

✔ Polynomial

- Already in descending order.
- Degree: 4
- Leading coefficient: 1
- Number of terms: 5

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8. `30 + y³ + y - 6y²`

✔ Polynomial

- Rearrange: `y³ - 6y² + y + 30`
- Degree: 3
- Leading coefficient: 1
- Number of terms: 4

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9. `17a - 11a² - 25 + 2a³`

✔ Polynomial

- Rearrange: `2a³ - 11a² + 17a - 25`
- Degree: 3
- Leading coefficient: 2
- Number of terms: 4

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10. `-3`

✔ Polynomial — constant polynomial!

- Degree: 0 (constant term has degree 0)
- Leading coefficient: -3
- Number of terms: 1

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## ✔ Final Answers Table:

| GIVEN | DEGREE | LEADING COEFFICIENT | NUMBER OF TERMS |
|-------------------------------|--------|----------------------|------------------|
| `2a³ + 3a⁴ + 18a + a⁵ - 9` | 5 | 1 | 5 |
| `√(3a²) - 3a + 1` | — | — | — |
| `3a³ - a⁻² + 2a - 1` | — | — | — |
| `1/a² - 3a³ + 4a - 3` | — | — | — |
| `x² - 9` | 2 | 1 | 2 |
| `11x + 10x² + 3` | 2 | 10 | 3 |
| `y⁴ - 2y³ - 13y² + 16y + 24` | 4 | 1 | 5 |
| `30 + y³ + y - 6y²` | 3 | 1 | 4 |
| `17a - 11a² - 25 + 2a³` | 3 | 2 | 4 |
| `-3` | 0 | -3 | 1 |

> For non-polynomials, leave the cells blank or write "N/A" or "Not a polynomial".

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## 🧠 Summary of Key Rules Used:

- Polynomials can only have non-negative integer exponents.
- No radicals over variables (like `√a` or `√(a²)` unless simplified to linear form — but still often excluded).
- No variables in denominators (like `1/x` or `x⁻¹`).
- Constant terms are polynomials of degree 0.
- Leading coefficient is the coefficient of the highest-degree term when written in standard form (descending powers).

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