Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

POLYNOMIAL OPERATIONS, Study notes of Calculus

POLYNOMIAL OPERATIONS. ADDITION AND SUBTRACTION: Adding and subtracting polynomials is the same as the procedure used in combining like terms.

Typology: Study notes

2021/2022

Uploaded on 08/05/2022

hal_s95
hal_s95 🇵🇭

4.4

(652)

10K documents

1 / 4

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
POLYNOMIAL OPERATIONS
ADDITION AND SUBTRACTION:
Adding and subtracting polynomials is the same as the procedure used in combining like terms.
When adding polynomials, simply drop the parenthesis and combine like terms. When subtracting
polynomials, distribute the negative first, then combine like terms.
Examples:
Addition:
2
3 73
4 10 2
3
3 4 7 10 5
17
Subtraction:
5
12 12
3 7 5
12 1 2
3 7 3
15 8
MULTIPLICATION:
1. Monomial times Monomial: To multiply a monomial times a monomial, just multiply the
numbers then multiply the variables using the rules for exponents.
Example:
2
5
2 · 5
· · ·
10
2. Monomial times Polynomial: Simply use the distributive property to multiply a monomial times a
polynomial.
Examples:
a. 2
3 8 2
23 28 2
6
16
b. 5
2
3 6 5
2
5
3 5
6 10
15
30
3. Binomial times a Binomial: To multiply two binomials, use the FOIL method (First times first,
Outside times outside, Inside times inside, and Last times last).
Example:
2 3 3 2 23
3 2 6
6
Special Products: The following formulas may be used in these special cases as a short cut to the
FOIL method.
Difference of Squares:

Example:
3 43 4 9
16
Perfect Squares:
2
Example:
4
24 4
8 16
2
Example:
3
23 3
6 9
pf3
pf4

Partial preview of the text

Download POLYNOMIAL OPERATIONS and more Study notes Calculus in PDF only on Docsity!

POLYNOMIAL OPERATIONS

ADDITION AND SUBTRACTION:

Adding and subtracting polynomials is the same as the procedure used in combining like terms. When adding polynomials, simply drop the parenthesis and combine like terms. When subtracting polynomials, distribute the negative first, then combine like terms.

Examples:

Addition:

䙦2ᡶ⡰^ ㎗ 3ᡶ ㎘ 7䙧 ㎗ 䙦3ᡶ⡰^ ㎘ 4ᡶ ㎘ 10䙧 㐄 2ᡶ⡰^ ㎗ 3ᡶ⡰^ ㎗ 3ᡶ ㎘ 4ᡶ ㎘ 7 ㎘ 10 㐄 5ᡶ⡰^ ㎘ ᡶ ㎘ 17

Subtraction:

䙦5ᡶ⡰^ ㎘ 12ᡶ ㎗ 1䙧 ㎘ 䙦2ᡶ⡰^ ㎗ 3ᡶ ㎘ 7䙧 㐄 5ᡶ⡰^ ㎘ 12ᡶ ㎗ 1 ㎘ 2ᡶ⡰^ ㎘ 3ᡶ ㎗ 7 㐄 3ᡶ⡰^ ㎘ 15ᡶ ㎗ 8

MULTIPLICATION:

  1. Monomial times Monomial: To multiply a monomial times a monomial, just multiply the numbers then multiply the variables using the rules for exponents.

Example: 䙦㎘2ᡶ⡰ᡷ䙧䙦5ᡶᡷ⡵䙧 㐄 ㎘2 · 5ᡶ⡰^ · ᡶ · ᡷ · ᡷ⡵^ 㐄 ㎘10ᡶ⡱ᡷ⡶

  1. Monomial times Polynomial: Simply use the distributive property to multiply a monomial times a polynomial.

Examples:

a. ㎘2ᡶ䙦ᡶ⡰^ ㎗ 3ᡶ ㎘ 8䙧 㐄 ㎘2ᡶ䙦ᡶ⡰䙧 ㎘ 2ᡶ䙦3ᡶ䙧 ㎘ 2ᡶ䙦㎘8䙧 㐄 ㎘2ᡶ⡱^ ㎗ 6ᡶ⡰^ ㎗ 16ᡶ b. 5ᡶ⡰䙦㎘2ᡶ⡲^ ㎗ 3ᡷ ㎘ 6䙧 㐄 5ᡶ⡰䙦㎘2ᡶ⡲䙧 ㎗ 5ᡶ⡰䙦3ᡷ䙧 ㎗ 5ᡶ⡰䙦㎘6䙧 㐄 ㎘10ᡶ⡴^ ㎗ 15ᡶ⡰ᡷ ㎘ 30ᡶ⡰

  1. Binomial times a Binomial: To multiply two binomials, use the FOIL method (First times first, Outside times outside, Inside times inside, and Last times last).

Example:

䙦ᡶ ㎗ 2䙧䙦ᡶ ㎘ 3䙧 㐄 ᡶ䙦ᡶ䙧 ㎗ ᡶ䙦㎘3䙧 ㎗ 2䙦ᡶ䙧 ㎗ 2䙦㎘3䙧 㐄 ᡶ⡰^ ㎘ 3ᡶ ㎗ 2ᡶ ㎘ 6 㐄 ᡶ⡰^ ㎘ ᡶ ㎘ 6

Special Products: The following formulas may be used in these special cases as a short cut to the FOIL method.

Difference of Squares: 䙦ᡓ ㎗ ᡔ䙧䙦ᡓ ㎘ ᡔ䙧 㐄 ᡓ⡰^ ㎘ ᡔ⡰

Example: 䙦3ᡶ ㎗ 4䙧䙦3ᡶ ㎘ 4䙧 㐄 9ᡶ⡰^ ㎘ 16 Perfect Squares: 䙦ᡓ ㎗ ᡔ䙧⡰^ 㐄 ᡓ⡰^ ㎗ 2ᡓᡔ ㎗ ᡔ⡰

Example: 䙦ᡶ ㎗ 4䙧⡰^ 㐄 ᡶ⡰^ ㎗ 2䙦ᡶ䙧䙦4䙧 ㎗ 4⡰^ 㐄 ᡶ⡰^ ㎗ 8ᡶ ㎗ 16

䙦ᡓ ㎘ ᡔ䙧⡰^ 㐄 ᡓ⡰^ ㎘ 2ᡓᡔ ㎗ ᡔ⡰

Example: 䙦ᡶ ㎘ 3䙧⡰^ 㐄 ᡶ⡰^ ㎘ 2䙦ᡶ䙧䙦3䙧 ㎗ 3⡰^ 㐄 ᡶ⡰^ ㎘ 6ᡶ ㎗ 9

  1. Polynomial times polynomial: To multiply two polynomials where at least one has more than two terms, distribute each term in the first polynomial to each term in the second.

Examples:

a. 䙦ᡶ⡰^ ㎗ 3ᡶ ㎘ 4䙧䙦ᡶ⡰^ ㎘ 6ᡶ ㎗ 5䙧 㐄 ᡶ⡰䙦ᡶ⡰䙧 ㎗ ᡶ⡰䙦㎘6ᡶ䙧 ㎗ ᡶ⡰䙦5䙧 ㎗ 3ᡶ䙦ᡶ⡰䙧 ㎗ 3ᡶ䙦㎘6ᡶ䙧 ㎗ 3ᡶ䙦5䙧 ㎘ 4䙦ᡶ⡰䙧 ㎘ 4䙦㎘6ᡶ䙧 ㎘ 4䙦5䙧 㐄 ᡶ⡲^ ㎘ 6ᡶ⡱^ ㎗ 5ᡶ⡰^ ㎗ 3ᡶ⡱^ ㎘ 18ᡶ⡰^ ㎗ 15ᡶ ㎘ 4ᡶ⡰^ ㎗ 24ᡶ ㎘ 20 㐄 ᡶ⡲^ ㎘ 3ᡶ⡱^ ㎘ 17ᡶ⡰^ ㎗ 39ᡶ ㎘ 20 b. 䙦2ᡶ ㎘ 3䙧䙦4ᡶ⡰^ ㎘ 5ᡶ ㎗ 1䙧 㐄 2ᡶ䙦4ᡶ⡰䙧 ㎗ 2ᡶ䙦㎘5ᡶ䙧 ㎗ 2ᡶ䙦1䙧 ㎘ 3䙦4ᡶ⡰䙧 ㎘ 3䙦㎘5ᡶ䙧 ㎘ 3䙦1䙧 㐄 8ᡶ⡱^ ㎘ 10ᡶ⡰^ ㎗ 2ᡶ ㎘ 12ᡶ⡰^ ㎗ 15ᡶ ㎘ 3 㐄 8ᡶ⡱^ ㎗ 22ᡶ⡰^ ㎗ 17ᡶ ㎘ 3 DIVISION:

  1. Division by Monomial: Each term of the polynomial is divided by the monomial and it is simplified as individual fractions.

Examples:

a. ⡱け

ㄘ⡹⡷け⡸⡩⡲ ⡱け 㐄^

⡱けㄘ ⡱け ㎘^

⡷け ⡱け ㎗^

⡩⡲ ⡱け 㐄 ᡶ ㎘ 3 ㎗^

⡩⡲ ⡱け

b. ⡩⡨げ

ㄘ⡹⡰⡳げ⡸⡰⡨ ⡹⡳げ 㐄^

⡩⡨げㄘ ⡹⡳げ ㎘^

⡰⡳げ ⡹⡳げ ㎗^

⡰⡨ ⡹⡳げ 㐄 ㎘2ᡷ ㎗ 5 ㎘^

⡲ げ

  1. Division by Binomial or Larger Polynomial:

Use the long division format as follows:

  • Both the divisor and the dividend must be written in descending order.
  • Any missing powers should be replaced by zero.
  • All remainders are in fraction form (remainder divisor⁄^ ) and are added to the quotient.

Examples:

a. 䙦ᡶ^2 ㎘ 2ᡶ ㎘ 15䙧 㐂 䙦ᡶ ㎗ 3䙧 㐄 ᡶ ㎘ 5

x x x x

b. ⡷け

ㄙ⡹け⡸⡱ ⡱け⡹⡰ 㐄 3ᡶ

⡱け⡹⡰

2 3 2

x x ㎘䙦ᡶ⡰^ ㎗ 3 ᡶ䙧 x x x x ㎘ 5 ᡶ ㎘ 15 ㎘䙦㎘ 5 ᡶ ㎘ 15 䙧 0

㎘䙦 9 ᡶ⡱^ ㎘ 6 ᡶ⡰䙧 6 ᡶ⡰^ ㎘ ᡶ ㎘䙦6ᡶ⡰^ ㎘ 4ᡶ䙧 3ᡶ ㎗ 3 ㎘䙦 3 ᡶ ㎘ 2 䙧 5

POLYNOMIAL OPERATIONS PRACTICE ANSWERS

1. 12ᡢ⡱^ ㎘ ᡢ ㎘ 5

2. 11ᡓ⡳^ ㎗ ᡓ ㎘ 6

3. 8ᡥ⡳^ ㎗ 9ᡥ

4. 12ᡥ⡳^ ㎗ 3ᡥ ㎘ 1

5. ㎘8ᡶ⡰^ ㎘ 6ᡶ ㎗ 6

6. 7ᡶ⡱^ ㎘ 7ᡶ ㎘ 2

7. 2ᡶ⡰^ ㎗ ᡶ ㎗ 2

8. ㎘4ᡶ⡰^ ㎗ 9ᡶ ㎘ 2

9. 16ᡶ⡰^ ㎘ 8ᡶ ㎗ 5

10. 6ᡶ⡰^ ㎗ 3ᡶ ㎗ 5

11. 5ᡶ⡱^ ㎘ 6ᡶ ㎗ 3

12. ㎘2ᡶ⡱^ ㎗ 10ᡶ⡰^ ㎘ 6ᡶ ㎗ 18

13. ㎘ᡶ⡱^ ㎗ 11ᡶ⡰^ ㎘ 8ᡶ ㎗ 10

14. ㎘2ᡶ⡱^ ㎗ 12ᡶ⡰^ ㎘ 8ᡶ ㎗ 10

19. 18ᡶ⡳^ ㎗ 10ᡶ⡱ᡷ

20. 10ᡶ⡲^ ㎗ 20ᡶ⡱ᡷ

21. 15ᡥ⡳^ ㎗ 25ᡥ⡲^ ㎘ 20ᡥ⡱^ ㎗ 30ᡥ⡰

22. ㎘4ᡶ⡲ᡷ ㎘ 28ᡶ⡱ᡷ⡰^ ㎗ 24ᡶ⡰ᡷ⡲

23. ᡶ⡰^ ㎗ 8ᡶ ㎗ 12

24. ᡶ⡰^ ㎗ 3ᡶ ㎘ 54

25. 12ᡶ⡰^ ㎘ 29ᡶ ㎗ 15

26. ᡶ⡰^ ㎘ 15ᡶ ㎗ 56

27. 30ᡓ⡰^ ㎗ 17ᡓ ㎗ 2

28. 10ᡶ⡰^ ㎗ 33ᡶᡷ ㎗ 20ᡷ⡰

29. 8ᡶ⡰^ ㎘ 14ᡶᡷ ㎘ 9ᡷ⡰

30. 36ᡰ⡰^ ㎘ 24ᡰ ㎘ 5

31. 36ᡕ⡰^ ㎘ 49

32. 9ᡶ⡰^ ㎗ 30ᡶᡷ ㎗ 25ᡷ⡰

33. ᡶ⡱^ ㎘ 3ᡶ⡰^ ㎗ 5ᡶ ㎘ 6

34. 10ᡶ⡱^ ㎘ 17ᡶ⡰^ ㎘ 6ᡶ ㎘ 35

39. ᡶ⡶^ ㎘ ⡳け

ㄣ ⡲ ㎘ 5ᡶ

  1. ㎘ᡶ⡰^ ㎗ ᡶ ㎗ (^) け⡵ㄘ ㎘ (^) け⡷ㄠ
  2. ᡶ ㎗ 2 ㎗ ⡴け
  3. ㎘ᡶ ㎗ 5 ㎘ (^) ⡱け⡳
  4. ᡶ⡶^ ㎘ ⡳け

ㄠ ⡰ ㎘ 5ᡶ

  1. ㎘2ᡶ⡰^ ㎗ 5ᡶ ㎗ (^) け⡷ㄘ ㎗ (^) け⡰ㄠ
  2. ᡘ⡰^ ㎘ 4ᡘ ㎗ 16
  3. 3ᡨ ㎘ 7 ㎗ (^) ぃ⡹⡩⡳
  4. 2ᡥ ㎘ 7 ㎗ (^) ぀⡸⡳⡱⡩
  5. ᡢ⡰^ ㎗ 4ᡢ ㎗ 16
  6. 4ᡨ ㎗ 3 ㎗ (^) ぃ⡹⡰⡩⡨
  7. 3ᡨ ㎘ 8 ㎗ (^) ぃ⡸⡲⡱⡩
  8. 4ᡶ ㎘ 1
  9. ㎘2ᡶ⡰^ ㎗ 2ᡶ ㎘ 1 ㎘ (^) け⡹⡩⡰
  10. 4ᡶ ㎘ 3
  11. 3ᡶ⡰^ ㎘ 3ᡶ ㎗ 1 ㎘ (^) ⡰け⡸⡩⡳