Prime Factorization of Monomials - Questions With Solutions

Review of Monomials and Prime Factorization

A monomial is the product of a real number and variables raised to non-negative integer powers.

Examples of monomials:   \( 2x \),   \( -5x^2 y \),   \( 30x^2 y^4 \)

What is prime factorization of monomials?

The prime factorization of a monomial 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. \( -30x^2y^3 = -2 \times 3 \times 5 \times x \times x \times y \times y \times y \)

Question 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. \( 2 \times 2 \times 2 \times 3 \times x \times x \times y \times y \)

Show Solution

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.

Question 2

What is the prime factorization of the following monomials?

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

Show Solution

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. \( 120x^2y^2 = 2 \times 2 \times 2 \times 3 \times 5 \times x \times x \times y \times y \)

Question 3

Find the prime factorizations of \( 5xy^2 \) and \( 20x^3y \) and then the prime factorization of \( 100x^4y^3 \) knowing that:

\[ 100x^4y^3 = (5xy^2) \times (20x^3y) \]

Show Solution

The prime factorization of each monomial is:

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. \( 100x^4y^3 = 2 \times 2 \times 5 \times 5 \times x \times x \times x \times x \times y \times y \times y \)