Intro to Polynomials Notes and Worksheets - Lindsay Bowden - Free Printable

Let's go through each problem step by step and solve them with explanations.

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1. How many terms does the polynomial have?

Polynomial: $ 7x^4 + 3x^2 - 10 $

- A term is a single part of a polynomial separated by addition or subtraction.
- Here, we have:
- $ 7x^4 $
- $ 3x^2 $
- $ -10 $

✔ Answer: 3 terms

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2. Classify the polynomial by degree and number of terms.

Polynomial: $ 8x^4 - 1 $

- Degree: The highest exponent of $ x $ is 4 → degree 4
- Number of terms: Two terms: $ 8x^4 $ and $ -1 $

Classification:
- Degree 4 → quartic
- 2 terms → binomial

✔ Answer: Quartic binomial

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3. What is the degree of the polynomial?

Polynomial: $ 3x^3 - 10x^2 + 17 $

- Degree = highest exponent of $ x $
- Exponents: 3, 2, 0 (constant term)
- Highest is 3

✔ Answer: 3

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4. Write the polynomial in standard form.

Polynomial: $ -18x^3 + 12x^5 + 7x^4 - 5x^2 + 14 $

- Standard form: Terms arranged in descending order of exponents.

List the terms by degree:
- $ 12x^5 $ (degree 5)
- $ 7x^4 $ (degree 4)
- $ -18x^3 $ (degree 3)
- $ -5x^2 $ (degree 2)
- $ 14 $ (degree 0)

✔ Answer: $ 12x^5 + 7x^4 - 18x^3 - 5x^2 + 14 $

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5. Classify the polynomial by degree and number of terms.

Polynomial: $ -6x^3 + 2x^2 + 8x - 5 $

- Degree: Highest exponent = 3 → cubic
- Number of terms: 4 terms

4 terms → quadrinomial

✔ Answer: Cubic quadrinomial

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6. Identify the degree and leading coefficient of the polynomial.

Polynomial: $ 2x^5 + 3x^2 + 10x $

- Degree: Highest exponent = 5
- Leading term: Term with highest degree = $ 2x^5 $
- Leading coefficient: Coefficient of that term = 2

✔ Answer: Degree = 5, Leading coefficient = 2

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7. Arrange the polynomial so it’s in standard form.

Polynomial: $ 13x^2 - 10x^4 + 5x^3 - 11 $

Order terms by descending degree:
- $ -10x^4 $ (degree 4)
- $ 5x^3 $ (degree 3)
- $ 13x^2 $ (degree 2)
- $ -11 $ (degree 0)

✔ Answer: $ -10x^4 + 5x^3 + 13x^2 - 11 $

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8. What is the leading coefficient of the polynomial?

Polynomial: $ -9x^4 + 5x^3 - 12 $

- Leading term = $ -9x^4 $ (highest degree)
- Leading coefficient = -9

✔ Answer: -9

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9. Classify the polynomial by degree and number of terms.

Polynomial: $ 7x^2 $

- Degree: 2 → quadratic
- Number of terms: 1 → monomial

✔ Answer: Quadratic monomial

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10. Rewrite the polynomial so that it’s in standard form.

Polynomial: $ 8x - 10x^3 + 2x^2 - 16 $

Arrange in descending powers of $ x $:

- $ -10x^3 $ (degree 3)
- $ 2x^2 $ (degree 2)
- $ 8x $ (degree 1)
- $ -16 $ (degree 0)

✔ Answer: $ -10x^3 + 2x^2 + 8x - 16 $

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✔ Final Answers Summary:

| # | Answer |
|---|--------|
| 1 | 3 terms |
| 2 | Quartic binomial |
| 3 | 3 |
| 4 | $ 12x^5 + 7x^4 - 18x^3 - 5x^2 + 14 $ |
| 5 | Cubic quadrinomial |
| 6 | Degree: 5, Leading coefficient: 2 |
| 7 | $ -10x^4 + 5x^3 + 13x^2 - 11 $ |
| 8 | -9 |
| 9 | Quadratic monomial |
| 10 | $ -10x^3 + 2x^2 + 8x - 16 $ |

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