Worksheet for practicing polynomial identification and analysis.

It looks like you've uploaded a practice worksheet titled *"Describing Polynomials"*, which includes questions about polynomial functions and their properties. Since I can't view images directly, I’ll help you solve the problems based on the text you provided.

Let’s go through each part of the worksheet step by step and solve it completely.

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Part A: Fill in the Blanks

1. An ________ degree polynomial must have at least one real zero.

- This is referring to the Fundamental Theorem of Algebra, but more specifically, odd-degree polynomials always cross the x-axis at least once because their end behaviors are opposite (one goes to +∞, the other to –∞).
- ✔ Answer: *Odd*

2. A polynomial function is written in ________. If its terms are written in descending order of exponents from left to right.

- When terms are arranged from highest to lowest exponent, it's called standard form.
- ✔ Answer: *Standard form*

3. The ________ is the number in front of the term with the highest exponent in the polynomial.

- This is the leading coefficient.
- ✔ Answer: *Leading coefficient*

4. A ________ has a polynomial with one term, a ________ has two terms, and a ________ has three terms.

- These are types of polynomials based on number of terms:
- One term → monomial
- Two terms → binomial
- Three terms → trinomial
- ✔ Answers: *Monomial, Binomial, Trinomial*

5. It is possible for an ________ degree polynomial to have no real zeros.

- Even-degree polynomials can open upward or downward and may never cross the x-axis (e.g., $ y = x^2 + 1 $).
- ✔ Answer: *Even*

6. The ________ is used to determine the end behavior of the graph of a polynomial function.

- The leading term (the term with the highest degree) determines the end behavior.
- ✔ Answer: *Leading term*

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Part B: Analyze Each Polynomial

We need to:
- Write each polynomial in standard form
- State the degree
- Identify the leading coefficient
- Classify by number of terms (monomial, binomial, trinomial)
- Determine end behavior using arrows

Let’s do this for each:

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#### 7. $ y = 7 - 2x $

- Standard Form: $ y = -2x + 7 $
- Degree: 1 (highest power of x is 1)
- Leading Coefficient: -2
- Classification: Binomial (two terms)
- End Behavior:
- As $ x \to -\infty $, $ y \to \infty $ → up on the left
- As $ x \to \infty $, $ y \to -\infty $ → down on the right
- Arrows:
↑ on left, ↓ on right

✔ Table Entry:
| Standard Form | Degree | Leading Coeff. | Classification | End Behavior |
|---------------|--------|----------------|----------------|--------------|
| $ y = -2x + 7 $ | 1 | -2 | Binomial | ↑ ↓ |

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#### 8. $ y = 3x^3 + x^2 - (-x^4 + x^2) $

Simplify:
$$
y = 3x^3 + x^2 + x^4 - x^2 = x^4 + 3x^3
$$
(Notice: $ -(-x^4 + x^2) = +x^4 - x^2 $; then combine like terms)

- Standard Form: $ y = x^4 + 3x^3 $
- Degree: 4
- Leading Coefficient: 1
- Classification: Binomial
- End Behavior:
- Even degree, positive leading coefficient → both ends go up
- Arrows: ↑ on left, ↑ on right

✔ Table Entry:
| Standard Form | Degree | Leading Coeff. | Classification | End Behavior |
|---------------|--------|----------------|----------------|--------------|
| $ y = x^4 + 3x^3 $ | 4 | 1 | Binomial | ↑ ↑ |

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#### 9. $ y = (2x)^2 + 3x - 1 $

First simplify:
$$
(2x)^2 = 4x^2
\Rightarrow y = 4x^2 + 3x - 1
$$

- Standard Form: $ y = 4x^2 + 3x - 1 $
- Degree: 2
- Leading Coefficient: 4
- Classification: Trinomial
- End Behavior:
- Even degree, positive leading coefficient → both ends up
- Arrows: ↑ on left, ↑ on right

✔ Table Entry:
| Standard Form | Degree | Leading Coeff. | Classification | End Behavior |
|---------------|--------|----------------|----------------|--------------|
| $ y = 4x^2 + 3x - 1 $ | 2 | 4 | Trinomial | ↑ ↑ |

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#### 10. $ y = (x + 2)^2 + 3 $

Expand:
$$
(x+2)^2 = x^2 + 4x + 4 \Rightarrow y = x^2 + 4x + 4 + 3 = x^2 + 4x + 7
$$

- Standard Form: $ y = x^2 + 4x + 7 $
- Degree: 2
- Leading Coefficient: 1
- Classification: Trinomial
- End Behavior: Both ends up → ↑ ↑

✔ Table Entry:
| Standard Form | Degree | Leading Coeff. | Classification | End Behavior |
|---------------|--------|----------------|----------------|--------------|
| $ y = x^2 + 4x + 7 $ | 2 | 1 | Trinomial | ↑ ↑ |

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#### 11. $ y = (2 + x)(2 - x) - 4 $

Use difference of squares:
$$
(2+x)(2-x) = 4 - x^2
\Rightarrow y = 4 - x^2 - 4 = -x^2
$$

- Standard Form: $ y = -x^2 $
- Degree: 2
- Leading Coefficient: -1
- Classification: Monomial
- End Behavior:
- Even degree, negative leading coefficient → both ends down
- Arrows: ↓ on left, ↓ on right

✔ Table Entry:
| Standard Form | Degree | Leading Coeff. | Classification | End Behavior |
|---------------|--------|----------------|----------------|--------------|
| $ y = -x^2 $ | 2 | -1 | Monomial | ↓ ↓ |

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#### 12. $ y = 3(x + 1)^2 - 3x^2 $

First expand:
$$
(x+1)^2 = x^2 + 2x + 1 \Rightarrow 3(x^2 + 2x + 1) = 3x^2 + 6x + 3
\Rightarrow y = 3x^2 + 6x + 3 - 3x^2 = 6x + 3
$$

- Standard Form: $ y = 6x + 3 $
- Degree: 1
- Leading Coefficient: 6
- Classification: Binomial
- End Behavior:
- Odd degree, positive leading coefficient → left down, right up
- Arrows: ↓ on left, ↑ on right

✔ Table Entry:
| Standard Form | Degree | Leading Coeff. | Classification | End Behavior |
|---------------|--------|----------------|----------------|--------------|
| $ y = 6x + 3 $ | 1 | 6 | Binomial | ↓ ↑ |

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#### 13. $ y = 2x - 2(x - 3) $

Distribute:
$$
2x - 2x + 6 = 6
$$

So $ y = 6 $

- Standard Form: $ y = 6 $
- Degree: 0 (constant)
- Leading Coefficient: 6
- Classification: Monomial
- End Behavior:
- Constant function → horizontal line
- No change in direction → flat
- Arrows: → → (horizontal)

But note: degree 0 polynomials are constants — they don’t have “end behavior” in the same sense as higher degrees, but we usually just write horizontal lines.

✔ Table Entry:
| Standard Form | Degree | Leading Coeff. | Classification | End Behavior |
|---------------|--------|----------------|----------------|--------------|
| $ y = 6 $ | 0 | 6 | Monomial | → → |

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✔ Final Completed Table:

| Example | Standard Form | Degree | Leading Coeff. | Classification | End Behavior |
|--------|----------------|--------|----------------|----------------|--------------|
| 7. $ y = 7 - 2x $ | $ y = -2x + 7 $ | 1 | -2 | Binomial | ↑ ↓ |
| 8. $ y = 3x^3 + x^2 - (-x^4 + x^2) $ | $ y = x^4 + 3x^3 $ | 4 | 1 | Binomial | ↑ ↑ |
| 9. $ y = (2x)^2 + 3x - 1 $ | $ y = 4x^2 + 3x - 1 $ | 2 | 4 | Trinomial | ↑ ↑ |
| 10. $ y = (x + 2)^2 + 3 $ | $ y = x^2 + 4x + 7 $ | 2 | 1 | Trinomial | ↑ ↑ |
| 11. $ y = (2 + x)(2 - x) - 4 $ | $ y = -x^2 $ | 2 | -1 | Monomial | ↓ ↓ |
| 12. $ y = 3(x + 1)^2 - 3x^2 $ | $ y = 6x + 3 $ | 1 | 6 | Binomial | ↓ ↑ |
| 13. $ y = 2x - 2(x - 3) $ | $ y = 6 $ | 0 | 6 | Monomial | → → |

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🔍 Summary Notes:

- Always simplify expressions before analyzing.
- Standard form means descending powers of x.
- Degree = highest exponent.
- Leading coefficient = coefficient of the leading term.
- End behavior depends on:
- Degree (even/odd)
- Sign of leading coefficient
- Even + positive → ↑ ↑
- Even + negative → ↓ ↓
- Odd + positive → ↓ ↑
- Odd + negative → ↑ ↓

Let me know if you'd like a visual version or explanation of any specific problem!