5.1&5.2 exponents 8 2 =8 8 = 642 4 = 2 2 2 2 = 16 x 2 = x xx 4 = x x x xbase = x exponent =...
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![Page 1: 5.1&5.2 Exponents 8 2 =8 8 = 642 4 = 2 2 2 2 = 16 x 2 = x xx 4 = x x x xBase = x Exponent = 2Exponent = 4 Exponents of 1Zero Exponents Anything to the](https://reader035.vdocument.in/reader035/viewer/2022062713/56649f4c5503460f94c6cd07/html5/thumbnails/1.jpg)
5.1&5.2 Exponents82 =8 • 8 = 64 24 = 2 • 2 • 2 • 2 = 16
x2 = x • x x4 = x • x • x • x Base = x Base = xExponent = 2 Exponent = 4
Exponents of 1 Zero ExponentsAnything to the 1 power is itself Anything to the zero power = 1
51 = 5 x1 = x (xy)1 = xy 50 = 1 x0 = 1 (xy)0 = 1
Negative Exponents
5-2 = 1/(52) = 1/25 x-2 = 1/(x2) xy-3 = x/(y3) (xy)-3 = 1/(xy)3 = 1/(x3y3)
a-n = 1/an 1/a-n = an a-n/a-m = am/an
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Powers with Base 10100 = 1101 = 10102 = 100103 = 1000104 = 10000
The exponent is the same as the The exponent is the same as the numbernumber of 0’s after the 1. of digits after the decimal where 1 is placed
100 = 110-1 = 1/101 = 1/10 = .110-2 = 1/102 = 1/100 = .0110-3 = 1/103 = 1/1000 = .00110-4 = 1/104 = 1/10000 = .0001
Scientific Notation uses the concept of powers with base 10.
Scientific Notation is of the form: __. ______ x 10(** Note: Only 1 digit to the left of the decimal)
You can change standard numbers to scientific notationYou can change scientific notation numbers to standard numbers
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Scientific NotationScientific Notation uses the concept of powers with base 10.
Scientific Notation is of the form: __. ______ x 10(** Note: Only 1 digit to the left of the decimal)
-25 321
Changing a number from scientific notation to standard formStep 1: Write the number down without the 10n part.Step 2: Find the decimal pointStep 3: Move the decimal point n places in the ‘number-line’ direction of the sign of the exponent.Step 4: Fillin any ‘empty moving spaces’ with 0.
Changing a number from standard form to scientific notationStep1: Locate the decimal point.Step 2: Move the decimal point so there is 1 digit to the left of the decimal.Step 3: Write new number adding a x 10n where n is the # of digits moved left adding a x10-n where n is the #digits moved right
5.321
.05321
.0 5 3 2 1= 5.321 x 10-2
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Raising Quotients to Powers
a n
b = an
bna -n b
= a-n
b-n= bn
an= b n
a
Examples: 3 2 32 94 42 16= =
2x 3 (2x)3 8x3
y y3 y3= =
2x -3 (2x)-3 1 y3 y3
y y-3 y-3(2x)3 (2x)3 8x3= = = =
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Product Ruleam • an = a(m+n)
x3 • x5 = xxx • xxxxx = x8
x-3 • x5 = xxxxx = x2 = x2
xxx 1
x4 y3 x-3 y6 = xxxx•yyy•yyyyyy = xy9 xxx
3x2 y4 x-5 • 7x = 3xxyyyy • 7x = 21x-2 y4 = 21y4
xxxxx x2
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Quotient Rule
am = a(m-n)
an
43 = 4 • 4 • 4 = 41 = 4 43 = 64 = 8 = 442 4 • 4 42 16 2
x5 = xxxxx = x3 x5 = x(5-2) = x3
x2 xx x2
15x2y3 = 15 xx yyy = 3y2 15x2y3 = 3 • x -2 • y2 = 3y2 5x4y 5 xxxx y x2 5x4y x2
3a-2 b5 = 3 bbbbb bbb = b8 3a-2 b5 = a(-2-4)b(5-(-3)) = a-6 b8 = b8
9a4b-3 9aaaa aa 3a6 9a4b-3 3 3 3a6
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Powers to Powers
(am)n = amn
(a2)3 a2 • a2 • a2 = aa aa aa = a6
(24)-2 = 1 = 1 = 1 = 1/256 (24)2 24 • 24 16 • 16 28 256
(x3)-2 = x –6 = x 10 = x4
(x -5)2 x –10 x 6
(24)-2 = 2-8 = 1 = 1
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Products to Powers
(ab)n = anbn
(6y)2 = 62y2 = 36y2
(2a2b-3)2 = 22a4b-6 = 4a4 = a4(ab3)3 4a3b9 4a3b9b6 b15
What about this problem?
5.2 x 1014 = 5.2/3.8 x 109 1.37 x 109
3.8 x 105
Do you know how to do exponents on the calculator?
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Square Roots & Cube Roots
A number b is a square root of a number a if b2 = a
25 = 5 since 52 = 25
Notice that 25 breaks down into 5 • 5So, 25 = 5 • 5
See a ‘group of 2’ -> bring it outside theradical (square root sign).
Example: 200 = 2 • 100 = 2 • 10 • 10 = 10 2
A number b is a cube root of a number a if b3 = a
8 = 2 since 23 = 8
Notice that 8 breaks down into 2 • 2 • 2 So, 8 = 2 • 2 • 2
See a ‘group of 3’ –> bring it outsidethe radical (the cube root sign)
Example: 200 = 2 • 100 = 2 • 10 • 10 = 2 • 5 • 2 • 5 • 2
= 2 • 2 • 2 • 5 • 5 = 2 25
3
3
3 3
3
3
3
3
Note: -25 is not a real number since nonumber multiplied by itself will be negative
Note: -8 IS a real number (-2) since-2 • -2 • -2 = -8
3
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5.3 PolynomialsTERM • a number: 5 • a variable X
• a product of numbers and variables raised to powers 5x2 y3 p x(-1/2)y-2 z
MONOMIAL-- Terms in which the variables have only nonnegative integer exponents.
-4 5y x2 5x2z6 -xy7 6xy3
A coefficient is the numeric constant in a monomial.
DEGREE of a Monomial– The sum of the exponents of the variables. A constant term has a degree of 0 (unless the term is 0, then degree is undefined).
DEGREE of a Polynomial is the highest monomial degree of the polynomial.
POLYNOMIAL - A Monomial or a Sum of Monomials: 4x2 + 5xy – y2 (3 Terms)Binomial – A polynomial with 2 Terms (X + 5)Trinomial – A polynomial with 3 Terms
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Adding & Subtracting Polynomials
Combine Like Terms
(2x2 –3x +7) + (3x2 + 4x – 2) = 5x2 + x + 5
(5x2 –6x + 1) – (-5x2 + 3x – 5) = (5x2 –6x + 1) + (5x2 - 3x + 5) = 10x2 – 9x + 6
Types of Polynomialsf(x) = 3 Degree 0 Constant Functionf(x) = 5x –3 Degree 1 Linear f(x) = x2 –2x –1 Degree 2 Quadraticf(x) = 3x3 + 2x2 – 6 Degree 3 Cubic
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5.4 Multiplication of Polynomials
Step 1: Using the distributive property, multiply every term in the 1st polynomial by every term in the 2nd polynomial
Step 2: Combine Like TermsStep 3: Place in Decreasing Order of Exponent
4x2 (2x3 + 10x2 – 2x – 5) = 8x5 + 40x4 –8x3 –20x2
(x + 5) (2x3 + 10x2 – 2x – 5) = 2x4 + 10x3 – 2x2 – 5x + 10x3 + 50x2 – 10x – 25
= 2x4 + 20x3 + 48x2 –15x -25
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Another Method for Multiplication
Multiply: (x + 5) (2x3 + 10x2 – 2x – 5)
2x3 10x2 – 2x – 5
x
5
2x4 10x3 -2x2 -5x
10x3 50x2 -10x -25
Answer: 2x4 + 20x3 +48x2 –15x -25
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Binomial Multiplication with FOIL
(2x + 3) (x - 7)
F. O. I. L.(First) (Outside) (Inside) (Last)
(2x)(x) (2x)(-7) (3)(x) (3)(-7)
2x2 -14x 3x -21
2x2 -14x + 3x -21
2x2 - 11x -21
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5.5 & 5.6: Review: Factoring Polynomials
To factor a polynomial, follow a similar process.
Factor: 3x4 – 9x3 +12x2
3x2 (x2 – 3x + 4)
To factor a number such as 10, find out ‘what times what’ = 10
10 = 5(2)
Another Example:Factor 2x(x + 1) + 3 (x + 1)
(x + 1)(2x + 3)
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Solving Polynomial Equations By Factoring
Solve the Equation: 2x2 + x = 0
Step 1: Factor x (2x + 1) = 0
Step 2: Zero Product x = 0 or 2x + 1 = 0
Step 3: Solve for X x = 0 or x = - ½
Zero Product Property : If AB = 0 then A = 0 or B = 0
Question: Why are there 2 values for x???
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Factoring Trinomials
To factor a trinomial means to find 2 binomials whose productgives you the trinomial back again.
Consider the expression: x2 – 7x + 10
(x – 5) (x – 2)The factored form is:
Using FOIL, you can multiply the 2 binomials andsee that the product gives you the original trinomial expression.
How to find the factors of a trinomial:
Step 1: Write down 2 parentheses pairs.Step 2: Do the FIRSTSStep3 : Do the SIGNSStep4: Generate factor pairs for LASTSStep5: Use trial and error and check with FOIL
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Practice
Factor:
1. y2 + 7y –30 4. –15a2 –70a + 120
2. 10x2 +3x –18 5. 3m4 + 6m3 –27m2
1. 8k2 + 34k +35 6. x2 + 10x + 25
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5.7 Special Types of FactoringSquare Minus a Square
A2 – B2 = (A + B) (A – B)
Cube minus Cube and Cube plus a Cube
(A3 – B3) = (A – B) (A2 + AB + B2)
(A3 + B3) = (A + B) (A2 - AB + B2)
Perfect Squares
A2 + 2AB + B2 = (A + B)2
A2 – 2AB + B2 = (A – B)2
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5.8 Solving Quadratic Equations General Form of Quadratic Equation
ax2 + bx + c = 0 a, b, c are real numbers & a 0
A quadratic Equation: x2 – 7x + 10 = 0 a = _____ b = _____ c = ______
Methods & Tools for Solving Quadratic Equations• Factor • Apply zero product principle (If AB = 0 then A = 0 or B = 0)• Quadratic Formula (We will do this one later)
Example1: Example 2:x2 – 7x + 10 = 0 4x2 – 2x = 0(x – 5) (x – 2) = 0 2x (2x –1) = 0x – 5 = 0 or x – 2 = 0 2x=0 or 2x-1=0 + 5 + 5 + 2 + 2 2 2 +1 +1
2x=1x = 5 or x = 2 x = 0 or x=1/2
1 -7 10
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Solving Higher Degree Equations
x3 = 4x
x3 - 4x = 0x (x2 – 4) = 0x (x – 2)(x + 2) = 0
x = 0 x – 2 = 0 x + 2 = 0
x = 2 x = -2
2x3 + 2x2 - 12x = 0
2x (x2 + x – 6) = 0
2x (x + 3) (x – 2) = 0
2x = 0 or x + 3 = 0 or x – 2 = 0
x = 0 or x = -3 or x = 2
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Solving By Grouping
x3 – 5x2 – x + 5 = 0
(x3 – 5x2) + (-x + 5) = 0
x2 (x – 5) – 1 (x – 5) = 0
(x – 5)(x2 – 1) = 0
(x – 5)(x – 1) (x + 1) = 0
x – 5 = 0 or x - 1 = 0 or x + 1 = 0
x = 5 or x = 1 or x = -1
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Pythagorean Theorem
Right Angle – An angle with a measure of 90°
Right Triangle – A triangle that has a right angle in its interior.
Legs
Hypotenuse
C A
B
a
b
cPythagorean Theorem
a2 + b2 = c2
(Leg1)2 + (Leg2)2 = (Hypotenuse)2