ACT Math · Lesson 1 of 5
Numbers, fractions, and factors
Keep fraction operations exact, and use factors to simplify before you expand or guess.
Fractions and rational expressions
To add fractions, use a common denominator. 1/3 + 1/4 = 4/12 + 3/12 = 7/12. Multiplying the denominators is one way to find a common multiple, but a smaller common denominator is allowed and often safer.
A rational expression is a fraction that contains variables. You may cancel a common factor only when it is a factor of the entire numerator and the entire denominator. Canceling a term that is added, such as the 2 in (x + 2)/(x + 2y), is not valid.
Factors and multiples
A factor divides a number with no remainder. The prime factorization of 36 is 2² × 3². The greatest common factor of two numbers is the product of the primes they share. The least common multiple includes the highest power of every prime that appears.
Factoring a polynomial reverses multiplication. x² − 9 = (x − 3)(x + 3). That identity is useful for simplifying, solving, or finding excluded values, but it is not the same as solving x² − 9 = 0 until you set the expression equal to zero.
Worked example
Simplify (x² − 9)/(x² − x − 6) for x ≠ 3 and x ≠ −2.
- Factor the numerator: (x − 3)(x + 3).
- Factor the denominator: (x − 3)(x + 2).
- Cancel the common factor x − 3, leaving (x + 3)/(x + 2). The original expression is still undefined at x = 3 and x = −2.
Why this works. Cancel factors, not terms, and keep the inputs that were never allowed.
Check your understanding
- Add fractions only after writing a common denominator.
- List prime factors when comparing a greatest common factor and a least common multiple.
- State the values excluded by an original denominator.