Ray's New Elementary Algebra · Arts. 87–93 · Unit 2: Chapter II: Theorems, factoring, etc.
12. Divisors, prime factors and common monomial factors
Goal: Your student breaks numbers and monomials into prime factors and takes out the largest monomial factor of a polynomial.
You'll need
- The book, open to the page
- Pencil and plenty of paper (or a whiteboard)
- A card to cover printed answers
What's in the book
Divisors, prime and composite numbers, the rule for prime factors of a number, prime and composite quantities, prime factors of a monomial, and separating a polynomial into a monomial times a polynomial. Answers are printed.
The lesson, step by step
Warm-up
Ask: What numbers divide 12 exactly? (1, 2, 3, 4, 6, 12.) Which of them are prime?
Divisors and primes
Read Arts. 87 and 88.
A Divisor or Measure of a quantity is any quantity that divides it without a remainder.
A Prime Number is one which has no divisors except itself and unity.
Note that Ray counts 1 among the primes ("The prime numbers are 1, 2, 3, 5, 7, ..."). Most books today do not; just mention it.
Prime factors of a number
Read the rule:
1. Divide by any prime number that will exactly divide it; divide the quotient again in the same manner, and so continue.
2. The last quotient and the several divisors will be the prime factors.
Work together:
1. What are the prime factors of 105 ? Ans. 3, 5, 7.
Then your student does 210 and 4290 (exercises 2 and 3).
Monomials
Read Arts. 89 to 92.
Resolve the coefficient into its prime factors. To these annex the literal factors.
Show: 15a²bc = 3 × 5 × a × a × b × c. Your student does the other two exercises.
Taking out a common factor
Read Art. 93.
Divide the given quantity by the greatest monomial that will exactly divide each of its terms; the divisor will be one factor and the quotient the other.
Guided practice:
1. x+ax Ans. x(1+a).
4. 4x²+6xy Ans. 2x(2x+3y).
Independent practice
Art. 93 exercises 2, 3, 5, 6 and 7. Check each by multiplying the factors back together.
Check
Your student can confirm one afterwards on the site's calculator (/homeschool/calculator/) with factor(4*x^2+6*x*y), after doing it by hand.
Wrap-up
Ask: Why is 2x the greatest common monomial factor of 4x² + 6xy, and not just 2 or just x?
Tip
Always multiply back. It takes ten seconds and catches nearly every slip.
Answers to the book's problems
The book prints the answer after each exercise.
Worked out for this site from the scan. If a number in your copy differs, trust the book. Every answer is in the Key ↗.