Factoring the Expression 10a³b² - 5a²b + 15a²b³

Factors of the Expression

Expression Analysis

• The given expression is:
  10a^3b^2 - 5a^2b + 15a^2b^3

Step 1: Identify Common Factors

• Identify the common numerical and variable factors in each term of the expression.

Numerical Coefficients

• Consider the coefficients of each term:
  • First term: 10
  • Second term: -5
  • Third term: 15

Greatest Common Factor (GCF) of Numerical Coefficients

• The GCF of 10, -5, and 15 is:
  • Factor breakdown:
  • 10 = 2 × 5
  • -5 = -1 × 5
  • 15 = 3 × 5
  • Thus, GCF is: 5

Variable Parts

• Now consider the variable components in each term:
  • First term: a^3b^2
  • Second term: -5a^2b
  • Third term: 15a^2b^3

Identify Common Variables:

• For variable a:

  • The lowest power in the terms:
  • First term: 3
  • Second term: 2
  • Third term: 2
  • Thus, the common factor is: a^2

• For variable b:

  • The lowest power in the terms:
  • First term: 2
  • Second term: 1
  • Third term: 3
  • Thus, the common factor is: b

Step 2: Combine the Common Factors

• Combine the numerical GCF and the variable common factors:
  • Combining yields: 5a^2b

Step 3: Factor the Expression

• To factor the expression, divide each term by the common factor found:
  • First term:
    10a^3b^2
    ightarrow rac{10a^3b^2}{5a^2b} = 2a^{3-2}b^{2-1} = 2a^1b^1 = 2ab
  • Second term:
    -5a^2b
    ightarrow rac{-5a^2b}{5a^2b} = -1
  • Third term:
    15a^2b^3
    ightarrow rac{15a^2b^3}{5a^2b} = 3b^{3-1} = 3b^2

Resulting Expression

• Therefore, factoring the entire expression gives:
  10a^3b^2 - 5a^2b + 15a^2b^3 = 5a^2b(2ab - 1 + 3b^2)

Summary of Factorization

• The fully factored form of the original expression is:
  5a^2b(2ab + 3b^2 - 1)