Ray's New Elementary Algebra · Arts. 195–197 · Unit 6: Chapter VI: Powers, roots and radicals
41. Square roots of monomials and polynomials
Goal: Your student takes square roots of monomials (with ±) and of polynomials that are perfect squares.
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
The rule for monomials, the double sign, imaginary quantities, then the rule for polynomials with two worked examples and exercises 3 to 12, and remarks on when a binomial or trinomial can be a square. Answers are printed.
The lesson, step by step
Warm-up
Square 3ab², 2x − 3y and x + 2.
Monomials
Read Art. 195.
Extract the square root of the coefficient, and divide the exponent of each letter by 2.
SayEvery positive quantity has two square roots, + and −. So the root of 4a²x² is ±2ax. A negative quantity has no square root; such roots are called imaginary.
Guided practice: monomials
Your student does exercises 1 to 6 of Art. 195 while you watch.
Polynomials
Read Art. 196 and the rule (steps 1 to 5). Work the book's example 2 together: 4x⁴ − 12x³y + 25x²y² − 24xy³ + 16y⁴ has root 2x² − 3xy + 4y².
Guided practice: polynomials
3. x²+4x+4 Ans. x+2.
6. x⁴+4x³+6x²+4x+1 Ans. x²+2x+1.
Your student works both by the rule, even though the first can be seen at a glance.
Independent practice
Exercises 4, 5 and 7 to 12. Square each answer to check. Afterwards your student may confirm one on the site's calculator with expand((x^2+2*x+1)^2).
When is a trinomial a square?
Read Art. 197. Test: is 9x² + 12xy + 16y² a perfect square? (No: the middle term would have to be 24xy.)
Wrap-up
Use the General Review on p. 176 for its second half.
Tip
The polynomial method copies the number method exactly. Show the two side by side if your student is lost.
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 ↗.