UNIT 9 -- POLYNOMIALS AND EXPONENTS
ADDING/SUBTRACTING POLYNOMIALS (Day 1)
Polynomial: The sum of monomials.
- Monomial: ____ term: ex. ______
- Binomial: ____ unlike terms:ex. ______
- Trinomial: ______unlike terms: ex. ______
Like Terms: Terms with the same ______to the same ______.
Ex 1:Ex 2:Ex 3:
Simplest form: When the polynomial contains no like terms.
5x3 + 8x2 + 2x3 + 7
DEGREE of a Monomial: The sum of the exponents of the variable symbols that appear
in the monomial. EX: For 6a2b3, Degree is ______.
DEGREE of a Polynomial: ______
Standard Form: ______
Constant Term: ______
Leading Coefficient: ______
Standard Form / Degree / Leading Coefficient3x2 + x5 – 7x
5x3 + 7 – x6
7 – 2x + 4x3
-9x3 + 2x – x2 + 1
Examples:
Find each sum or difference by combining the parts that are alike. Make sure simplified expressions are in STANDARD FORM:
- (4x2 + x + 7) + (2x2 + 3x + 1)2. (3x3 – x2 + 8) – (x3 + 5x2 + 4x – 7)
Practice:
Simplify the following polynomials:
1.10y – (3y + 6) 2.(–4a + 6b) + (3a – b)
3.–5m + 6n + 8p – (6n + 3m) 4.(2x + 4x2 – 7) – (x2 + 7 – 8x)
5.(a2 + 3b2 – 4a) + (2a2 – 6b2 )
NOTE: Whatever follows the word FROM is the expression that goes 1st when subtracting!
1.Subtract 9a – 3b from 4a – 7b
2.From 2x3 – 4x2 + x, subtract 8x3 + 2x2 – 3x
Let’s try some different types of problems…
1.Describe and correct the error(s) in the following problems.
(a)(b)
2.The perimeter of a triangle can be represented by the expression 3x2 – 7x + 2.
Write a polynomial that represents the measure of the third side.
MULTIPLYING WITH EXPONENTS (DAY 2)
POWERS OF THE SAME BASE
Exponent Rule for Multiplying with the same Base:xa xb = x
You must have the same base before exponents are ______!
Practice:
1.x5x4 = ______6.c (c5) = ______
2.103102 = ______7.24252 = ______
3.b6(b)= ______8.(q2)(2q4) = ______
4.m4am3a= ______9.(9w2x8)(w6x4) = ______
5.z3z4z5= ______10.(14fg2h2)(-3f4g2h2) = ______
11.Write expressions for the areas of the two rectangles, separately, in the figures given
below:
FINDING THE POWER OF A POWER
Exponent Raised to an Exponent Rule:(xa)c = x
The base remains the same, and the exponents are ______together.
Option 1:Separate EACH piece of theBASE and raise it to the indicated Exponent, then simplify.
Option 2: Write the expression the number of times indicated by Exponent #, then simplify.
Simplify the following:
1.(a4)2 = ______8.(523)4 = ______
2.(y6)3 = ______9.(2232)4= ______
3.(2a4)2 = ______10.(xy3)2= ______
4.(x4)3 = ______11.(2432)3= ______
5.(x3)4 = ______12. = ______
6.(rs)3= ______13.______
7.(x3y4)2 = ______14.(2a3)4(a3)5 = ______
DISTRIBUTION (DAY 3)
As we saw earlier this year, the distributive property may be used to simplify expressions:
1)-5(4m – 6n)2)-8(2x2 – 3x – 5)3)(12x – 4w)
Multiplying Monomials with Variables by Polynomials
We will now take it a step further to include monomials with variables to be distributed over polynomial expressions. Multiply Coefficients AND Use your EXPONENT RULES from yesterday!
Examples:Use the distributive property to write each of the following expressions as the sum of monomials (IN OTHER WORDS, SIMPLIFY).
1)4x(5x + 6)2)(d2 – 3d)5d3)-3ab(5a2 – 7b2)
Practice:
1)3a(4 + a)2)x(x + 2) + 13)(-5w – 3)w2
Recall the two rectangles from Day 2 Notes? What if we put them together? Now, write an expression for the area of this new rectangle:
Multiplying Polynomials with Polynomials
What if you had to distribute a binomial to a binomial? Or a binomial to a trinomial?
Apply the same concept of multiplying, but do it more than once! Take turns!
Example 1:Use the distributive property to simplify:
(x + 4)(x + 5)[Show Arrows]
This can be rewritten as: x(x + 5) + 4(x + 5)
Let’s finish the problem:
Example 2:Use the distributive property to simplify:(x – 1)(x2 – x + 1)
Practice:
1)(x + 2)(x + 3)2)(x – 4)(x – 2)3)(2x + 5)(3x – 4)
4)(2z – 1)(3z2 + 1) 5)z(2z + 1)(3z – 2)
6)(x – 1)(x2 – x + 1)7)(b + 2)(b2 – 3b + 7)
8)(y2 + 2y – 1)(y + 1)9) The length of a rectangle is represented by 5y2 – 7 and the
width is represented by 3y3.
Determine the area of the
rectangle. [A picture may help!]
DISTRIBUTION (CONTINUED) – DAY 4
SPECIAL CASES:
Case 1: Binomials that have the same letters and variables in the same order, but different middle signs are called ______. They are unique because when you distribute them, the middle terms cancel each other out.
1. (x + 7)(x – 7)2.(4x + 3)(4x – 3)
Simplify the following:
3.(10w – 1)(10w + 1)4.(3 + d)(3 – d)
Case 2: When squaring terms in parentheses you must square the ENTIRE term, which means that you must WRITE ITTWICE, then distribute!!!
(2x + 3)2 =
Simplify:
1.(a + 1)22.(5 + b)2
3.(3x – 2)24.(a + b)2
5.Describe and correct the error shown here:
6.Simplify:=
7.The length of one side of a square is x – y. Express the area of the square as a trinomial.
8.You are creating a frame to surround a rectangular picture. The width of the frame around the picture is the same on every side, as shown.
Write a polynomial that represents the total area of the picture and the frame.
Find the combined area of the picture and the frame when the width of the frame is 4 inches.
9.You are designing a rectangular skateboard park on a lot that is on the corner of a city block. The park will have a walkway along two sides. The dimensions of the lot and the walkway are shown in the diagram.
Write a polynomial that represents the area of the skateboard park (which includes the walkway).
What is the area of the park (with the walkway) if the walkway is 3ft wide?
DIVIDING WITH EXPONENTS (DAY 5)
x5 x2 can also be written as This really means = =
Using the Division Rule: VS. Canceling Method,
Simplify the following:
1.x9 x5 = ____2.y5 y = ____3. c5 c5 = ____
4.105 103 = ____5.3a 32c= ____6. x5a x2a =____
7.y10b y2b = ____8. = ______
9. = ______10. = ______
11. = ______12. = ______
[Oops: Neg. Exponent for #11! See next page!]
What if the exponent on the top is smaller than the exponent on the bottom?
Ex:
Negative Exponents:
Basic Formula: x–n =
Simplify the following with positive exponents only.
1.2. 3. 4. 5.
6. 7. 8. 9. 10.
What would happen if we ended up with a Zero Exponent?
A BASE TO THE ZERO EXPONENT WILL EQUAL 1!!!
1.2.3.-4(x2)04.
5. 6. 7.
DIVIDING (Continued) -- DAY 6
DIVIDING A MONOMIAL BY A MONOMIAL (from yesterday)
Procedure:
- Divide the coefficients
- Divide powers with same base by ______EXPONENTS
Examples:
1.2.
DIVIDING A POLYNOMIAL BY A MONOMIAL
STEPS:
- Do the “Pretzel” Method and Rewrite the problem to show separatefractions.
2. Perform division for each fraction by dividing coefficients first, then, dividing
variables (by subtracting exponents if terms have the same base).
3. Make sure answer is simplified.
Simplify:
1.2.
3.4.
5.6.
7.8.
9.The volume of a rectangular prism (box) is represented by . The height of the box is 6ft and the width is x feet. Find the length of the box in terms of x. [RECALL that V = lwh] DRAW A PIC!
10.The blueprint of a sandbox on a playground shows that it has an area of . Using the picture below determine the width of the sandbox expressed as a binomial in terms of x.
11.A soup can holds a volume represented by . The radius of the soup can is represented by . Using the volume of a cylinder formula, , find the height of the cylinder in terms of x. DRAW A PIC!
12.A swimming pool holds a volume represented by . If the depth of the pool is represented by and the width is represented by , find the length of the pool expressed as a binomial in terms of x. [Use Volume formula from #9 above; depth is like height]. DRAW A PIC!
1