Prime Factorization of Monomials - Questions With Solutions

What is prime factorization of monomials? Grade 11 examples and questions are presented along with Solutions and explanations included.

Review of Monomials and Prime Factorization

A monomial is the product of a real number and variables raised to non negative integer powers. Example of monomials: \( 2x \), \( \; -5x^2 y \), \( \; 30x^2 y^4 \)

What is prime factorization of monomials?

The prime factorization of a monomials is obtained by writing the whole number in the monomial in prime factorization followed by the product of the variables.

Examples of monomials in prime factorization form

  1. \( 10x = 2 \times 5 \times x \)
  2. \( 20x^2 = 2 \times 2 \times 5 \times x \times x \)
  3. \( -30x^2y^3 = -2 \times 3 \times 5 \times x \times x \times y \times y \times y \)

Questions

  1. Which of the following is not a prime factorization?
    1. \( 2 \times 10 \times x \)
    2. \( 2 \times 7 \times x \times x \)
    3. \( 4 \times 4 \times 4 \times x \times x \times y \times y \times y \)
    4. \( 2 \times 2 \times 2 \times 3 \times x \times x \times y \times y \)
  2. What is the prime factorization of the following monomials?
    1. \( 28xy^2 \)
    2. \( 32x^3y \)
    3. \( 70x^3y^3 \)
    4. \( 120x^2y^2 \)
  3. Find the prime factorizations of \( 5xy^2 \) and \( 20x^3y \) and then the prime factorization of \( 100x^4y^3 \) knowing that \[ 100x^4y^3 = (5xy^2) \times (20x^3y) \]

Solutions to the Above Questions

  1. Solution to Question 1

    1. The 10 in \( 2 \times 10 \times x \) is not a prime number and therefore \( 2 \times 10 \times x \) is not a prime factorization.
    2. \( 2 \times 7 \times x \times x \) is a prime factorization.
    3. The 4 in \( 4 \times 4 \times 4 \times x \times x \times y \) is not a prime number and therefore \( 4 \times 4 \times 4 \times x \times x \times y \) is not a prime factorization.
    4. \( 2 \times 2 \times 2 \times 3 \times x \times x \times y \times y \) is a prime factorization.

  2. Solution to Question 2

    1. \( 28xy^2 = 2 \times 2 \times 7 \times x \times y \times y \)
    2. \( 32x^3y = 2 \times 2 \times 2 \times 2 \times 2 \times x \times x \times x \times y \)
    3. \( 70x^3y^3 = 2 \times 5 \times 7 \times x \times x \times x \times y \times y \times y \)
    4. \( 120x^2y^2 = 2 \times 2 \times 2 \times 3 \times 5 \times x \times x \times y \times y \)

  3. Solution to Question 3

    The prime factorization of each monomial.
    1. \( 5xy^2 = 5 \times x \times y \times y \)
    2. \( 20x^3y = 2 \times 2 \times 5 \times x \times x \times x \times y \)
    3. \( 100x^4y^3 = 2 \times 2 \times 5 \times 5 \times x \times x \times x \times x \times y \times y \times y \)

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