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UNIT 4
POLYNOMIALS
Math 11 Unit 4 Introduction p. 1 of 1
A. Algebraic Skills
Unit 4 Polynomials Introduction
Problem:
Derrek has a part time job changing tires. He gets paid the same
amount for each tire he changes. Here is a list of the number of tires
he changed for three days:
Monday:
6 tires
Tuesday:
8 tires
Wednesday: 10 tires
How could you express algebraically, the total amount of money he
earned for the three days?
Let x = the amount of money earned per tire
Solution: 6x + 8x + 10x
= 24x
If Derrek earns $5.00 per tire, how much does he make?
Solution: 24(5) = $120. Derrek earns $120 over the 3 days.
A polynomial is an expression that uses numbers and variables
(letters) to simplify a math problem. The terms in a polynomial can
be added, subtracted, multiplied, and divided in order to simplify the
expression.
Math 11 Unit 4 Objectives p. 1 of 1
A. Algebraic Skills
Unit 4 Polynomials Objectives
• To become familiar with polynomial terminology.
• To add and subtract polynomials.
• To multiply a monomial by a monomial.
• To multiply a polynomial by a monomial.
• To divide a polynomial by a monomial.
Math 11 Unit 4 Outline p. 1 of 1
A. Algebraic Skills
Unit 4 Polynomials
1. Polynomial Terminology
Exercise 1
2. Adding and Subtracting Polynomials
Exercise 2
3. Multiplication of Monomials by Monomials
Exercise 3
4. Multiplication of Polynomials by Monomials
Exercise 4
5. Division of Polynomials by Monomials
Exercise 5
Unit 4 Review
Unit 4 Exam
Math 11 Unit 4 Lesson 1 p. 1 of 2
Math 11
Unit 4
Lesson 1
SWBAT become familiar with polynomial terminology
1. Polynomials
Term:
- basic element of all algebraic expressions
- terms are separated by a + or – sign
- has 2 components:
1. Numerical Coefficient (number)
2. Literal Coefficient (variable/letter and exponent)
Term
e.g.
17x3
Numerical
Coefficient
Literal
Coefficient
Constant:
a term with no variables; a number
e.g. -5
Polynomial:
an algebraic expression containing one or more terms.
Monomial:
one term
e.g. 7x3
Binomial:
two terms
e.g. 8x2 + 2y
Trinomial:
three terms
e.g. 10x2 + 6xy + 3y
Degree:
is the greatest exponent total of any term in a polynomial
Standard Form:
when a polynomial is written in descending (large to small) order
of degrees (exponents) from left to right
Math 11 Unit 4 Lesson 1 p. 2 of 2
State the degree for the following polynomials and what type they are:
Ex. 1. 2a2b6c3
Degree: 2 + 6 + 3 = 11
Type: monomial
Ex. 2. 7k3 – 3r2 + 5kr
3
2
2 Degree: choose highest = 3
Type: trinomial
Ex. 3. –9x4y7 + 17xy9
11
10 Degree: 11
Type: binomial
Arrange the terms in descending order of degrees.
Ex. 4. 5b – 5b4 – 9b2 – 7 – b3
= -5b4 – b3 – 9b2 + 5b – 7
Ex. 5. 3x2y + 2y2 – 7 – x3 in terms of x
= -x3 + 3x2y + 2y2 - 7
Exercise 1
Math 11 Unit 4 Exercise 1 p. 1 of 2
Math 11 Unit 4
Exercise 1
Fill in the blanks using the following words. Some words may be used twice.
exponent
letter
term
number
one
two
three
four
monomial
binomial
trinomial
polynomial
ascending
descending
1. A _________ is a basic element of all algebraic expressions.
2. A degree is the greatest ______________ of any term in the polynomial.
3. A ______________ has two terms.
4. A numerical coefficient is the same as a _____________ .
5. A polynomial is written in standard form when the terms are in ___________ order of
degrees from left to right.
6. There is/are ___________ term(s) in the polynomial 3xy + 7x – 8yz.
7. A literal coefficient is the same as a __________________.
8. The degree of the polynomial 2abcd is ____________.
9. A ________________ has one term.
10. A term has ______ components.
11. There is/are ________ term(s) in the polynomial 9a + 5b.
12. A trinomial has _______ terms.
13. A _________________ is an algebraic expression containing one or more terms.
14. There is/are ___________ term(s) in the polynomial 3xy.
15. The coefficient in the expression, 4st, is the _____________.
Math 11 Unit 4 Exercise 1 p. 2 of 2
State the degree for the following polynomials and what type they are:
16. 8a2 + 5a – 3
19. 6a6 - 9a3b2
22. 15a9 + 7a5
17. 4gh
20. 4g4h5j + 2g2h3j2 + 3g6j2
23. 5r2t2 + 8r3t3
18. 9 – 7u
21. 7y2 + 8y + 9
24. 9a5b2 + 2a3b5
25. 7gh3j10
Arrange in descending order of degrees:
26. 5a + 4a4 – a7 + 2
31. 14y5 + 3y + 12y4
27. 10d3 + d – 3 +d4 – d2
32. 9t13 – 9t15 + 3 – t10
28. –5t2 – 9t6 + 3 – t4
33. 5a5 + 9a2 – 4a12 + 2a6
29. 4xy4 + 5x3y – 2x5 + 7 in terms of x
34. 6xy2 + 12x2y – 3x6 + 7x8 in terms of x
30. a5b – ab7 + a4b2 – 6 + a3b8 in terms of a
35. 11f3 + 2f – 5f10 +6f4 – 9f2 + f7 – 7f12
Math 11 Unit 4 Lesson 2 p. 1 of 2
Math 11
Unit 4
Lesson 2
SWBAT add and subtract polynomials.
2. Adding and Subtracting Polynomials
•
Like Terms:
You can only add and subtract like terms
same literal coefficient (same variable and exponent)
Examples of like terms:
o 7x2, -8x2, 13x2
o 12abc, -abc, 3abc
o 5r2t, 10r2t, -2r2t
•
like term is x2
like term is abc
like term is r2t
Add or subtract the coefficient (number), but keep the variable (letter) the
same.
Add or subtract like terms:
Ex. 1. 15a + 7a
= 22a
Ex. 2. 9w – 3w + 2w
= 8w
Ex. 3. 7x – (-6y) – 3x + 2y
= 7x – 3x + 6y + 2y
{Group like terms}
= 4x + 8y
Ex. 4. 4ab + 7ab – 1 + 2ab – 3ab + 8
= 4ab + 7ab + 2ab – 3ab – 1 + 8
= 10ab + 7
Math 11 Unit 4 Lesson 2 p. 2 of 2
* When subtracting a polynomial in brackets, remember to subtract all terms inside the
brackets.
Ex. 5. (6b – 3c) - (-8b + 9c)
= 6b – 3c + 8b - 9c
= 6b + 8b – 3c - 9c
= 14b - 12c
Ex. 6. (4x + 9) – (2x – 5)
= 4x + 9 – 2x + 5
= 4x – 2x + 9 + 5
= 2x + 14
Exercise 2
Math 11 Unit 4 Exercise 2 p. 1 of 2
Math 11 Unit 4
Exercise 2
Add or subtract like terms.
1. 5a + 3a
10. 6b + (-3b) – 7b
2. 5k + 8k
11. 8h – (-7h)
3. 8n – 12n
12. 11x – 4y + 9y – 16x
4. 4w – 7w
13. 7p – 8r – 5p – 3r
5. 8s – 13s
14. 9b + 9a + 2b – 2a + 7b
6. 5y – 9y – 2y
15. 6h + 6 + 5h – 13 + 7 – 15h
7. 4t – 5t + 9t
16. -4mn + 6pq + 2mn – 5pq
8. -5n + 9n + 16n + 9n
17. 17rs – (-2wx) – 6rs + 2wx
9. 2a – (-6a)
18. 4y – 10c + 6y – 2c +(-4y) + c
Math 11 Unit 4 Exercise 2 p. 2 of 2
19. (12a – 3c) + (3a + 4c)
25. (7r – 3p) – (-6r + 9p)
20. (7y – 2) + (2y – 17)
26. (x + 9) + (-2x – 7)
21. (11k – 4m) – (2k – 15m)
27. (6x – 17) – (-7x – 2)
22. (-7y – 2) + (2y + 17)
28. (4a + b) – (6a – b)
23. (11k – 4m) + (-2k – 15m)
29. (-2y + 18) – (-2y + 17)
24. (4x + 8y) + (x – 10y)
30. (7x + y) – (7x – y)
Math 11 Unit 4 Lesson 3 p. 1 of 1
Math 11
Unit 4
Lesson 3
SWBAT multiply a monomial by a monomial.
3. Multiplication of Monomials by Monomials
Steps:
1. Multiply the coefficients (the numbers)
2. Write down the common bases (letters)
3. Add the exponents of the common bases
Multiply:
Ex. 1. –5(9p3)
= -45p3
Ex. 2. 3x3 × 7x6
= (3 × 7) (x3 × x6)
= 21x9
Ex. 3. (3m3n4)(-7m5n)
= -21m8n5
Ex. 4. 10s2t3 x 3st4
= 30s3t7
Ex. 5. –3a2bc × 5b3c2 × -2a3b4
= 30a5b8c3
Exercise 3
Math 11 Unit 4 Exercise 3 p. 1 of 2
Math 11 Unit 4
Exercise 3
Multiply.
1. -6(3y)
2. -4(-8a)
3. 3c × 8c
4. -5x × 4x
5. 8w • 9w
6. (-a)(-12a5)
7. 6t3 × 8t6
8. -2p × 25p4
9. 3s5 × 6s3
10. 4m2 × 11m8
11. c3d × c2d3
12. 10a6b × 5a8b12
13. (-3x2y4)(8xy3)
14. 7h4k2 • 8hk3
Math 11 Unit 4 Exercise 3 p. 2 of 2
15. -12k6m × -k3m
16. (4xy5)(-7xy)
17. 3w4x2y • 9wx3y5
18. (-6m2p3r2)(-6m2pr2)
19. 2y5 • 6y • 3y7
20. a7 • a3 • a2
21. (3y)(12y2)(y2)
22. (-7c3)(-2c)(-5c4)
23. (a)(3b3)(-8ab5)
24. (-5)(a2y5)(12a6y)
25. (-9ax)(-3a2x3)(-4x)
26. (4r4)(-7p7r3)(-pr2)
27. (m5r2)(3m)(-2r2p7)
28. (-14a2)(-2b3c)(-a3bc2)
29. (4pr2)(5p2r3)
30. (-2xy2)(-x2y)
Math 11 Unit 4 Lesson 4 p. 1 of 1
Math 11
Unit 4
Lesson 4
SWBAT multiply a polynomial by a monomial.
4. Multiplication of Polynomials by Monomials
We use the distributive property to multiply. Multiply the monomial on the outside of the
brackets by each term inside the brackets.
Multiply:
Ex. 1. 5(2x3 + 3x)
= (5 × 2x3) + (5 × 3x)
= 10x3 + 15x
Ex. 2. -2r(7r3 – 10r6)
= (-2r ×7r3) – (-2r ×10r6)
= -14r4 + 20r7
Ex. 3. 3x2(2x2 + 4x – 2)
= 6x4 + 12x3 - 6x2
Ex. 4. 3s2t2(10s2t3 + 3st4 – 2st)
= 30s4t5 + 9s3t6 - 6s3t3
Exercise 4
Math 11 Unit 4 Exercise 4 p. 1 of 2
Math 11 Unit 4
Exercise 4
Multiply.
1. 7(2y – 5)
2. f(f2 + 3f)
3. 8z(z2 – 3z)
4. 6d(3d3 + d2)
5. 2n2(10n – 7)
6. v4(8v2 + 5v)
7. 2xy(9x2y3 – x)
8. cd3(3cd + 4c4)
9. 11ab(3a3b2 + 11)
10. 6(3y3 – 9y2 – 5y)
11. n(5n4 – n2 + 4)
12. 3p(4p3 + 5p2 - 1)
13. t3(-3t3 + t2 - 7)
Math 11 Unit 4 Exercise 4 p. 2 of 2
14. 5w4(w4 – w2 + 2w)
15. cdf(c2 + cd + cdf)
16. p2r3(2pr + 4p2 – 3r)
17. 9ab(3a2b + 2ab2 - 1)
18. -3(t3 – 2t2 – 7t + 5)
19. 6k(2k3 – k2 + 7k - 1)
20. –z3(2z3 + 5z2 + 3z + 4)
21. -3h(4h4 + 2h3 – 6h2 – 6h)
22. -xy(3xz + 2y - y2 – 3yz)
23. a3b(ab4 + a2b – b2 + b)
24. -2k2(-9k5 - 6k3 - 1)
25. -2ab2(-12a4b3 + 3abc – b2 + a2b)
Math 11 Unit 4 Lesson 5 p. 1 of 1
Math 11
Unit 4
Lesson 5
SWBAT divide a polynomial by a monomial.
5. Division of Polynomials by Monomials
Divide each term of the polynomial by the monomial. Remember to subtract exponents
of like bases.
Divide:
Ex. 1.
12v 2
4v
= 3v
75a 5
Ex. 2.
−25a
= -3a4
Ex. 3. 48c7d3 ÷ 8c5d3
= 6c2
Ex. 4.
18 x 3 + 24 x 5
6x
=
18 x 3 24 x 5
+
6x
6x
= 3x2 + 4x4
21 f 6 − 18 f
Ex. 5.
3f 2
4
21 f 6 18 f 4
=
−
3f 2
3f 2
= 7f4 – 6f2
Exercise 5
Math 11 Unit 4 Exercise 5 p. 1 of 2
Math 11 Unit 4
Exercise 5
Divide.
35m 2
1.
5m
10. 16rp3 ÷ 2rp2
11. 25n4 ÷ 5n2
2.
3.
24w
−3
72a 3
8a
12. 24p3 ÷ (-12p2 )
13. -36t10 ÷ t4
14. -60w12 ÷ 15w10
4.
100h 7
20h 3
15. 6c2d ÷ 2cd
5.
36w 8
− 12 w
16.
20 x + 5
5
6.
81a 4
− 9a 2
17.
6m + 32
2
n4m5
7. 3 2
n m
18.
10w + 28
2
19.
25a + 5
5
20.
55a − 11b
11
8.
64h 5 k 2
4hk
9. 80m8r10 ÷ 10mr8
Math 11 Unit 4 Exercise 5 p. 2 of 2
21.
24d + 72
8
22.
− 18 y 3 − 24 y
−6
23.
− 36d 5 + 8d 2
4d
24.
8 x 6 + 12 x 4
4x3
25.
14n 2 − 7n
7n
26.
4a 5 + 16a 3
4a 2
27.
54 f 6 − 27 f 3
9f 3
28.
40 g 12 − 64 g 10
− 8g 5
29.
21w 5 x 3 − 14w 2 x 2
7w2 x
30.
32 x 2 y 6 − 40 x 5 y 4
− 8 xy 2
Math 11 Unit 4 Review p. 1 of 4
Name: _________________________________
Date: _________________________
Math 11 Unit 4
Review
What is the type and degree of the following polynomials:
1. 8d3 + 7d2
2. 10t5u3
3. 3a2b3c2
4. -8x4y5 + 9y7 – 3xy4
5. -6x6y3z
6. 4n2m3 – 4nm2
7. -7x2 – 12x2y2 + 4y3
Write the following polynomials in descending order:
8. 9x3 + 4x7 – 9x - 2
9. 10f2 + 12f8 -3f7 + f – 14f5 + 9
10. -3d5 + 7 – 4d6 + 9d2
Math 11 Unit 4 Review p. 2 of 4
Add or subtract the following:
11. 4w + 7w
12. 11uw + (-12uw) + 4uw
13. -12y + 5y + 17y + (-9y)
14. 7x + 4 + 8x + 6
15. 5p – 7 – 2p
16. 3x – 5 – 9x + 6
17. 5m – 6n + 2p – m – 5n + 3p
18. 4m – 11m – 27 + 7m + 15
19. 3p2 – 7p + 8 + 3p – 4p2
20. (2n + 4) – (n – 5)
21. (3x – 5y) – (7x + 2y)
Math 11 Unit 4 Review p. 3 of 4
Multiply the following:
22. 6m6 × 3m4
23. (5z2)(4z3)
24. 4m2 × 11m8
25. -12k6m × -k3m
26. (3x)(-x4)(-2x2)
27. (-7c3)(-2c)(-5c4)
28. 8(x - 4)
29. -5(2r + 3)
30. 2m(5m + 3)
31. -5d2(2d + 3)
32. f2(3f3 + 5)
33. 3j3(4j3 – 7j2)
34. 2p(4p3 + 5p2 – 1)
Math 11 Unit 4 Review p. 4 of 4
Divide the following:
35.
36 w 8
− 12 w 5
36.
24 p 6
12 p 4
37. 99h5i8 ÷ 11h5i7
38.
42 y 2 − 56
14
39.
− 18 y 3 + 6 y 2
6y
40.
64 z 3 − 40 z 2
− 8z