algebra 1 unit 6a – polynomials and exponents...applications of exponents 1. find the volume of...
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Algebra 1
Unit 6A – Polynomials and Exponents
Monday Tuesday Wednesday Thursday Friday
26 A Day 27 B Day 28 A Day 29 B Day 30 A Day
Law of Exponents
− Multiplication
− Power to Power
− Divide
− Negative
− Zero
Combination of All Exponent Properties
− incorporate all the laws together
− applications (geometry formulas)
Retest – CBA #4
Adding/Subtracting
Polynomials
Multiplying a
Monomial and a
Polynomial
Quiz – Multiplying
and Dividing
Exponents
Feb. 2 B Day 3 A Day 4 B Day 5 A Day 6 B Day
Adding/Subtracting
Polynomials
Multiplying a
Monomial and a
Polynomial
Quiz – Multiplying
and Dividing
Exponents
Multiplying Binomials
− multiplying monomial and binomial
− multiplying a binomial and a binomial
− box method
− algebra tiles Elaboration Day
9 A Day 10 B Day 11 A Day 12 B Day 13 A Day
Test – Exponents and Polynomials
Start Unit 6B – Factoring
1
2
EEEExxxxpppplllloooorrrreeee NNNNooootttteeeessss
ACTIVITY 1ACTIVITY 1ACTIVITY 1ACTIVITY 1
1. Write out x3 in expanded form. Next, to this, write out x
5 in expanded form with a multiplication dot between
them. This expanded form represents x3 ⋅ x
5.
2. Now, write this expanded form using only one base.
Therefore, x3x
5 =
3. Write out x12
x9 in expanded form.
4. Now, write this expanded form using only one base.
Therefore, x12
⋅ x9 =
Using this knowledge, simplify x30
x25
without writing in expanded form.
Algebraic Definition: xa ⋅ x
b =
Verbal Definition: When multiplying like bases…,
ACTIVITY 2ACTIVITY 2ACTIVITY 2ACTIVITY 2
1. Write out 3x3 ⋅ 2x
2 in expanded form.
2. Next, use the commutative property to group the coefficients and the like bases together.
3. Lastly, multiply the coefficients with each other. Then, rewrite the variables using one base.
Thus, 3x3 ⋅ 2x
2 =
4. Write out –2x7 ⋅ –3x
5 ⋅ 4x in expanded form.
5. Next, use the commutative property to group the coefficients and the like bases together.
6. Now, simplify and write this back into exponent form.
Therefore,–2x7 ⋅ –3x
5 ⋅ 4x =
Using this knowledge, simplify (–4x10
)(5x)( –2x) without writing in expanded form.
Algebraic Definition: Fxa ⋅ Gx
b =
Verbal Definition: When multiplying like bases with leading
coefficients…,
3
ACTIVITY 3ACTIVITY 3ACTIVITY 3ACTIVITY 3
1. Write x3 ⋅ y
4 in expanded form.
2. Use the commutative property to group like bases. Then, rewrite back into one exponent form.
3. What do you notice about x3 ⋅ y
4?
4. Write 2x3 ⋅ 4y
5 in expanded form.
5. Use commutative property to group the coefficients and the like bases.
6. Now simplify and write in exponent form.
Therefore, 2x3 ⋅ 4y
5 =
Using this knowledge, simplify (–8x10
)(6y3) without writing in expanded form.
Algebraic Definition: Fxa ⋅ Gy
b =
Verbal Definition: When multiplying terms with unlike
bases…
ACTIVITY 4ACTIVITY 4ACTIVITY 4ACTIVITY 4
1. Write (3x3)
2 in expanded form. Make sure you write all terms in expanded form. You may have to expand twice.
2. Use the commutative property to group the coefficients and the like bases.
3. After grouping the like bases and coefficients, use your new knowledge of multiplication and rewrite in simplified
exponent form.
Therefore, (3x3)
2 =
4. Write out (3x2y
3)
3 in expanded form. Make sure you write all terms in expanded form. You may have to expand twice.
5. Use the commutative property to group the coefficients and the like bases.
6. Use your new knowledge of multiplication and rewrite in simplified exponent form.
7. What relationship do you notice about the exponents inside the parentheses and outside the parentheses?
8. What relationship do you notice between the coefficient inside the parentheses and the exponent outside the
parentheses?
Using this knowledge, simplify (3x10
y8)
4 without writing in expanded form.
Algebraic Definition: (Fxa)b =
Verbal Definition: When raising a power to a power…
4
Using Exponent Properties
Simplify each expression.
1. 3
39
2km
m14k 2. (4x
2y)(2xy
2) 3. x
2(3xy)(xy
4)
4. 52
25
b9a
b3a 5.
34
65
mk
m3k- 6. (6x
2)(2x)
3
7. (-4x3y)(x
2y
2)(y) 8.
35
22
m16k
m4k 9. 3(x
2y)
2(xy
2)4
10. 52
64
yx4
yx16−
− 11.
95
71
ba20
ba40−−
−−
12. (2mn)0
13. 0
00
)dc(
dc
+
+ 14.
12
7
k
k−
−
15.
3
32
82
vt12
vt4
−
−
Find the missing factor.
16. -3u4v
2 = (u
2v)( ) 17. 32uv
5 = (-16v
2)( )
18. 27x4y
3 = (9x
4y)( ) 19. 14x
9y
6 = (-7x
2y
6)( )
20. Find the product of -3a2b
4c(2ab – 3a
2bc + 1).
Name Date
5
Triangle Square Rectangle Cylinder
Cube Rectangular Prism Triangle Rectangle
6
Applications of Exponents
1. Find the volume of the rectangular prism.
2. Simplify the following expression:
3 3 5
2 3 9
26a b c
65a b c
3. Find the area of a triangle with a base of 6x3y
5 and a height of -3x
2y
3.
4. Simplify the following expression:
( )
( )
32
33
2x
4x
5. If the area of a rectangle is 24x8y
3 and its length is 6x
2y, find the width of the rectangle.
6. Find the volume of a cube with sides 3x2y
3.
7. Find the area of the rectangle.
Name Date
2x
3x
x
9y-1
z4
22x9
7
8. Simplify the expression: (2a2)(3a
2) + (4a
3)(a)
9. Find the length of a rectangle.
10. Simplify the expression:
5 2
1
x y
z
−
−
11. Find the area of square with sides 3a5b.
12. Find the area of the triangle.
13. If the volume a rectangular prism is 100m3n
4, the width is 5m
2, and the length is 2mn
what is the height?
14. A cylinder has a radius of 2x2y and a height of 3x
3. What is the volume of the cylinder in
terms of π?
115yx4 yx52A
3=
35yx4
−
yx193
8
WARM-UP #_____
ALL of the problems below are incorrect. Explain and correct the errors. (There may be
more than one way to correct each problem.)
1. ( ) 1yx3yx2
yx6 02
0
5
43
=+
2. 4x0 = 0
3. (2x3)
4 = 8x
12
4. 4
242
y4
xyx4
−=−
−
5. x8x4
x122
3
=
6. 5.1
2
3
x3x4
x12=
7. 4x5 ⋅ 2x
3 = 6x
8
8. 4x5 ⋅ 2x
3 = 8x
11
9
10
Summary:
When you add or subtract polynomials, you add or subtract the coefficient of the bases with the same exponents. You
must combine Like Terms. Just remember to distribute the negative when you subtract Polynomials.
Adding and Subtracting Polynomials
Recall:
Monomials –
Binomials –
Trinomials –
Polynomials –
Coefficient –
Addition Rule: Combine the like terms
together.
1. (5a2 + a + 12) + (2a
2 – 3a – 10) =
2. (4a – 5) + (3a + 6) =
3. (3p2 – 2p + 5) + (p
2 – 7p + 7) =
4. (2a2 – ab + b
2) + (3a
2 + 5ab – 7ab
2) =
Subtraction rule: Combine the like terms
together. Make sure to distribute the negative.
1. (x2 + y
2) – (–x
2 + y
2) =
2. (10x2 + 5x – 6) – (8x
2 – 2x + 7) =
3. (2x2 + 5xy + 4y
2) – (2x
2 + 5xy + 4y
2) =
4. (x2 + 2xy + y
2) + (x
2 – xy – y
2) =
11
Multiplying a Monomial and a Polynomial
To multiply a polynomial and a monomial, distribute the monomial to each term in the
polynomial. Remember to:
• the coefficients, and then
• the exponents when multiplying variables with the same base.
1. x(7x2 + 4) 2. 4y(–y
3 – 2y – 1)
Sometimes when we have a polynomial it is easier to multiply using a box method.
3. 2x3(x
3 + 3x
2 – 2x + 5) 4. –5b
3(4b
5 – 2b
3 + b – 11)
5. Application: Write the polynomial
that represents the area of the
shaded region.
6. Distance (d), rate (r), and time (t)
are related by the formula d = rt. If
a ball rolls 36p4q
9 feet for 4p
2q
3
minutes, what is the rate?
2x
3x
10
8
12
Perimeter and Area
Find the area of each figure.
1. 2.
3. 4.
Find the perimeter of each figure.
5. 6.
7. 8.
Name Date
3x2y
6x3y – 4x + 2xy
4a5b
2
17a2 – 3ab + 5b
2
2xy3z
-2 5x
2 + 7xy – 3
2x
3x + 4
2x – 7
4x – 5
5x – 9
x – 6
x + 3
3x + 8
4x – 15
13
9. Find the perimeter of the equilateral triangle.
10. The sides of a triangle have lengths of (8x – 10), (2x – 5), and (x + 13). If the perimeter of the
triangle is 31 inches, find the length of each side of the triangle.
2x + 31
14
WARM-UP #_____
Simplify each expression.
1. (-3x2y)
3(11x
3y
5)
2 2.
2 25 4
2
2m n mn
4m 5n
•
3. Find the length of the missing side given the perimeter of the triangle.
P = 6x2 – 11x + 8
4. Find the area of a rectangle with sides (2x4y) and (-3xy
2 + xy – 7).
5. Use the pictures below and combine them using the Punnett square (box
method). Circle like terms.
?
x2 – x + 1
3x2 + 7x + 1
15
Algebra Tiles
1.
4. Draw the remaining tiles after
cancelling out zero pairs.
__________
Box 1
__________
Box 2 & 3
__________
Box 4
2. Use #1 to answer the questions
below.
a. If the figures represent algebra
tiles, what is the algebraic
expression for the tiles that are
at the top of the box?
b. What is the expression for the
tiles that are situated vertically?
5. Use the tiles in #4 and replace
the tiles with algebraic
expressions in the
corresponding box.
Circle like terms.
3. Use the information from #1 and #2.
Combine the tiles in each box by
multiplying and placing the new tile in
the box in the corresponding box
below.
Cancel out the zero pairs.
6. a) Simplify the expression from #5 by
combining like terms.
b) What do you notice about the
expressions for #4 and #6a?
1 2
3 4
16
Multiplying Binomials – Explain
Show the multiplication by drawing algebra tiles for each product.
1. 2x(x – 1) 2. (x – 3)(x + 5)
Result: Result:
Use the Box Method to find the product of the binomials.
3. (x + 5)(x – 3) 4. (x – 4)(x + 9)
Result: Result:
17
Continue using the Box Method to find the product of binomials.
5. (2x – 5)(3x + 4) 6. (x – 7)(8x – 9)
Result: Result:
7. (5x – 1)2 8. (x + 3)
2
Result: Result:
18
Multiplying Binomials
1. (x + 2)(x + 7) =
2. (z + 4)2 =
3. (a – 3)(a – 8) =
4. (x – 10)(x + 2) =
5. (3t + 2)(7t – 9) =
6. (2x – 3 )(2x + 3) =
7. (7y – 4)(2y + 5) =
8. (3d + 8)(3d – 8) =
9. (y + 6)(y – 3) =
10. (2u + 5h)(2u – 5h) =
Name Date
19
Polynomials and Exponents
Find each product.
1. (u + 3)2 2. (u – 8)
2 3. (2u + 5)
2
4. (1 – 4u)2 5. (u + 2v)
2 6. (7u – 3v)
2
7. (uv + 6)2 8. (u + v)(u – v)
Simplify each expression.
9. (-2x + 3x2) – (7x – 4) + (9x
2 – 3x) 10. (-5x
2 + 2x – 1) + (6x
3 + 2x
2 – 5)
11. (j3k
2)3 • (k
2)4 12. (-3x
6)2
13. (-7x5y
2)2
14. (-kd)2(-kd
2) 15.
4
05
x15
yx60 16. n
6 + n + n
6
S. 4u2 + 20u + 25
G. 4u2 + 16u + 25
A. u2 + 6u + 9
U. u2 + 4uv + 4v
2
D. 49u2 – 31uv + 9v
2
L. 16u2 – 8u + 1
E. u2 – 16u + 64
Q. u2v
2 + 12uv + 36
W. u2 + 7uv + 4v
2
M. 49u2 – 42uv + 9v
2
R. u2 – v
2
T. v2 – u
2
E. 16u2 + 8u + 1
Name Date
20
Some of the measures of polygons are given. P represents the measure of the perimeter. Find
the measure of the other side or sides.
17. P = 5x2 – 9x + 5 18. P = 16x
2 – 12
19. Which equation best represents the area, A, of the rectangle below?
A. A = 2x + 2(x + c)
B. A = x2 + (x + c)
2
C. A = x(x + c)
D. A = 2x(x + c)
20. The area of a rectangle is 144a8b
4 square units. If the width of the rectangle is 8a
2b
2 units,
what is the length in units?
A. 18a6b
2 units
B. 136a6b
2 units
C. 152a10
b6 units
D. 1152a10
b6 units
21. Tammy drew a floor plan for her kitchen, as shown below. Which expression represents
the area of Tammy’s kitchen floor in square units?
A. 6x2 + 30x + 5
B. 6x2 + 13x + 5
C. 10x + 12
D. 5x + 6
22. Find the area of a rectangle with a length of (2x + 7) inches and a width (5x – 1) inches.
?
x2 – x + 1
3x2 + 7x + 1
? (square)
x + c
x
21
WARM-UP #_______
1. Simplify each expression.
(4b – 3)(b – 7) (3a – 5) + (2a + 6) (5x + 2) – (x – 7)
2. Find the missing side with the given information.
a. P = 11x2 – 7x + 18 b. V = 196m
7n
6π
7m2n
3
?
5x2 – 3
?
2x2 – 5x + 9
x2 + 7
4x – 3
22
Review – Exponents and Polynomials
Simplify each of the following.
1. c5 • c
—9 • c 2. )gh6)(hg
2
1(
263−− 3. (wx)(xy)(wy)
4. 23
510
yx12
yx96− 5.
htam
hatm972
354
6. 93
58
unf45
unf18
7. sr63
sr146
32
−
− 8. (100a
10b
19c
6)0 9. (-2w
6x
2y
3)2
10. (3bc)2(4b
4)3 11. (y – 5)(2y + 3) 12. (4b – 3)(b – 7)
13. (5x – 8)(2x – 5) 14. (x – 7)2 15. (2r + 3)
2
16. -5g2h
3k(2gh – 3k
3 + g
2h
2k
2) 17. 3w
3(6w
2 – 4w + 1)
18. (2x2 – 6x + 9) + (3x
2 – 11 – 8x) 19. (7y – 4w + 10) + (8w + 3 – 6y)
20. (5y3 – 6y
2) – (7y
3 – 3y
2 + y) 21. (5b
4 – 8d
2) – (2 + 5b
4 – 4d
2)
Name Date
23
22. P = 13x + 8y 23. P = 5x2 – 9x + 5
? = _____________________ ? = _____________________
24. P = 16x2 – 12
? = _____________________
25. Find the area. 26. Find the length.
27. Find the area. 28. Find the perimeter.
x2 – x + 1
3x2 + 7x + 1
?
? 7x + 3y
4x + 4y
3b3c
4d
18bc6d
9
4x2y
5 A = 68x
5y
2
2x – 7
2x – 3
x + 7
5x + 9
x + 7
2x – 3
(square) ?
24