Skip to content

Exercises on Polynomial Identification

Below are some exercises to help practice identifying polynomials based on their terms, degree, coefficients, and form. The solutions are provided after each question for clarity.

Exercise 1: Identify the Degree and Type of Polynomial

For each of the following polynomials, identify the degree, the number of terms, and the type (monomial, binomial, trinomial, or general polynomial).

  1. \( P(x) = 5x^3 - 2x^2 + 7x - 1 \)
  2. \( P(x) = 3x + 4 \)
  3. \( P(x) = 7x^5 - 3x^2 + x \)
  4. \( P(x) = -2x^4 + 5x^3 - x^2 + 1 \)
  5. \( P(x) = 9x^6 \)

Solution 1:

  1. \( P(x) = 5x^3 - 2x^2 + 7x - 1 \)
  2. Degree: 3 (the highest exponent is 3 from \( 5x^3 \))
  3. Number of terms: 4 terms

  4. Type: General polynomial (since there are more than three terms)

  5. \( P(x) = 3x + 4 \)

  6. Degree: 1 (the highest exponent is 1 from \( 3x \))
  7. Number of terms: 2 terms

  8. Type: Binomial

  9. \( P(x) = 7x^5 - 3x^2 + x \)

  10. Degree: 5 (the highest exponent is 5 from \( 7x^5 \))
  11. Number of terms: 3 terms

  12. Type: Trinomial

  13. \( P(x) = -2x^4 + 5x^3 - x^2 + 1 \)

  14. Degree: 4 (the highest exponent is 4 from \( -2x^4 \))
  15. Number of terms: 4 terms

  16. Type: General polynomial

  17. \( P(x) = 9x^6 \)

  18. Degree: 6 (the highest exponent is 6 from \( 9x^6 \))
  19. Number of terms: 1 term
  20. Type: Monomial

Exercise 2: Identify the Leading Term and Leading Coefficient

For each of the following polynomials, identify the leading term and the leading coefficient.

  1. \( P(x) = 4x^2 + 3x + 2 \)
  2. \( P(x) = -5x^4 + x^3 - 2x \)
  3. \( P(x) = 7x^5 + x^2 + 4 \)
  4. \( P(x) = 2x - 9 \)
  5. \( P(x) = 10x^3 - 3x^2 + 7x - 1 \)

Solution 2:

  1. \( P(x) = 4x^2 + 3x + 2 \)
  2. Leading term: \( 4x^2 \) (the term with the highest degree)

  3. Leading coefficient: 4

  4. \( P(x) = -5x^4 + x^3 - 2x \)

  5. Leading term: \( -5x^4 \)

  6. Leading coefficient: -5

  7. \( P(x) = 7x^5 + x^2 + 4 \)

  8. Leading term: \( 7x^5 \)

  9. Leading coefficient: 7

  10. \( P(x) = 2x - 9 \)

  11. Leading term: \( 2x \)

  12. Leading coefficient: 2

  13. \( P(x) = 10x^3 - 3x^2 + 7x - 1 \)

  14. Leading term: \( 10x^3 \)
  15. Leading coefficient: 10

Exercise 3: Determine the Degree of Polynomials with Multiple Variables

For each of the following polynomials, identify the degree of the polynomial.

  1. \( P(x, y) = 4x^2y^3 + 5xy^2 - 3x^3 \)
  2. \( P(a, b) = a^3b^2 + ab + 1 \)
  3. \( P(x, y) = x^4 + 2x^2y^2 + y^4 \)
  4. \( P(m, n) = 3mn^2 - 5n + m^3n \)
  5. \( P(x, y) = 6x^3y + 4xy^3 \)

Solution 3:

  1. \( P(x, y) = 4x^2y^3 + 5xy^2 - 3x^3 \)

  2. Degree: 5 (the highest degree term is \( 4x^2y^3 \), where \( 2 + 3 = 5 \))

  3. \( P(a, b) = a^3b^2 + ab + 1 \)

  4. Degree: 5 (the highest degree term is \( a^3b^2 \), where \( 3 + 2 = 5 \))

  5. \( P(x, y) = x^4 + 2x^2y^2 + y^4 \)

  6. Degree: 4 (both \( x^4 \) and \( y^4 \) are degree 4 terms)

  7. \( P(m, n) = 3mn^2 - 5n + m^3n \)

  8. Degree: 4 (the highest degree term is \( m^3n \), where \( 3 + 1 = 4 \))

  9. \( P(x, y) = 6x^3y + 4xy^3 \)

  10. Degree: 4 (both \( 6x^3y \) and \( 4xy^3 \) have a total degree of 4)

Exercise 4: Classify Polynomials Based on Their Form

Classify the following polynomials as monomial, binomial, trinomial, or general polynomial based on the number of terms.

  1. \( P(x) = 2x^2 - 5 \)
  2. \( P(x) = 3x^3 - 2x + 1 \)
  3. \( P(x) = 4x \)
  4. \( P(x) = x^4 + x^3 + x^2 + x + 1 \)
  5. \( P(x) = 5x^2 + 3x - 2 \)

Solution 4:

  1. \( P(x) = 2x^2 - 5 \)

  2. Classification: Binomial (two terms)

  3. \( P(x) = 3x^3 - 2x + 1 \)

  4. Classification: Trinomial (three terms)

  5. \( P(x) = 4x \)

  6. Classification: Monomial (one term)

  7. \( P(x) = x^4 + x^3 + x^2 + x + 1 \)

  8. Classification: General polynomial (five terms)

  9. \( P(x) = 5x^2 + 3x - 2 \)

  10. Classification: Trinomial (three terms)

Exercise 5: Identify the Standard Form of Polynomials

Rewrite the following polynomials in standard form (terms in descending order of degree):

  1. \( P(x) = x - 4x^2 + 7x^3 \)
  2. \( P(x) = 5 - x^3 + 2x^2 \)
  3. \( P(x) = 2x + 3x^5 - 4x^3 \)
  4. \( P(x) = -3 + x + x^4 \)
  5. \( P(x) = -x + 2x^2 + x^5 \)

Solution 5:

  1. \( P(x) = x - 4x^2 + 7x^3 \)

  2. Standard form: \( 7x^3 - 4x^2 + x \)

  3. \( P(x) = 5 - x^3 + 2x^2 \)

  4. Standard form: \( -x^3 + 2x^2 + 5 \)

  5. \( P(x) = 2x + 3x^5 - 4x^3 \)

  6. Standard form: \( 3x^5 - 4x^3 + 2x \)

  7. **( P(x

) = -3 + x + x^4 )
- Standard form**: \( x^4 + x - 3 \)

  1. \( P(x) = -x + 2x^2 + x^5 \)
  2. Standard form: \( x^5 + 2x^2 - x \)

These exercises cover polynomial identification by focusing on the degree, number of terms, leading terms, and standard form. They also touch on the recognition of polynomials with multiple variables and different forms.