Download Interactive Study Guide for Students: Trigonometric Functions

Interactive Study Guide for Students
Chapter 5: Polynomials
Section 1: Monomials
Monomials
Examples
A __________________ is an expression that is a number, a variable,
or the product of a number and one or more variables. They cannot
contain variables in denominators, variables with exponents that are
negative, or variables under radical signs.
__________________ are monomials that contain no variables.
Simplify:
1. ( 3x3y2)(-4x2y4)
2. p3/p8
The number in front of any variable is called a ___________.
The ___________ of a monomial is the sum of the exponents of its
variables.
3. (a3)4
A _____________ is an expression of the form xn.
To _____________ an expression containing powers means to rewrite
the expression without any parentheses or negative exponents.
Negative Exponents:
Product of Powers:
Quotient of Powers:
a-n = ________
a n n n
) = a /b
b
5. ________
aman = am+n
________ = am-n
Properties of Powers: (am)n = amn
(
4.(-2p3s2)5
(
(ab)m = ambm
a -n b n
) =( )
b
a
or bn/an
Scientific Notation
The form that you usually write numbers in is ___________
____________________. A number is in ___________
_________when it is in the form a x 10n, where 1< a < 10 and n is an
integer.
Real-world problems often involve units on measure. Performing
operations with units is known as _____________
_________________.
Evaluate:
6. (4 x 105)(2 x 107)
7. (2.7 x 10-2)(3 x 106)
Chapter 5: Polynomials
Section 2: Polynomials
Add and Subtract Polynomials
Examples
A __________________ is a monomial or a sum of monomials. The
monomials that make up a polynomial are called ____________ of
the polynomial. ________ _________ are when the monomials have
the same variables. When a polynomial has two unlike terms, it is
called a __________.
Is it a polynomial? If so, state
the degree.
1. 1/6x3y5 – 9x4
2. x +
x +5
Subtract and simplify.
3. (3x2 -2x + 3)–(x2 + 4x –2)
Multiply Polynomials
Remember the __ __ __ __ method? It is used to multiply binomials.
__ irst, __uter, __nner, __ast
f
Multiply and simplify.
4. 2x(7x2 -3x + 5)
l
(x + a)(x + b)
i
5. (3y + 2)(5y + 4)
o
6. (n2 + 6n – 2)(n + 4)
Chapter 5: Polynomials
Section 3: Dividing Polynomials
Use Long Division
Examples
Use all of your skills when dividing, especially ones learned last
section.
1. 4x3y2 + 8xy2 – 12x2y3
4xy
2. (z2 + 2z – 24)  (z – 4)
3. (t2 + 3t – 9)(5 – t)-1
Use Synthetic Division
______________ ___________ is a simpler process for dividing a
polynomial by a binomial. You just use the coefficients and constants.
Step 1: Write the terms of the dividend so that the degrees of the
terms are in descending order. Then write just the coefficients as
shown at the right.
Step 2: Write the constant r of the divisor x – r to the left. Bring
down the first coefficient down.
Divide 5x3 - 13x2 + 10x -8 by
x-2
4.
5x3 - 13x2 + 10x - 8
2
5
-13
10
-8
________________________
Step 3: Multiply the first coefficient by r. Write the product under
the second coefficient. Then add the product and the second
coefficient.
5
Step 4: Continue multiplying the sum by r until you have used all of
the coefficients.
Step 5: If the final answer is 0, then it has divided evenly, and the
remaining equation is the bottom numbers. If it is not a 0, then that is 5. Divide (8x4 – 4x2 + x +4) by
the remainder.
(2x – 1)
Chapter 5: Polynomials
Section 4: Factoring Polynomials
Factoring Polynomials
Examples
1. Factor 6x2y2 - 2xy2+6x3y
Factoring Techniques:
# of terms
Technique
General Case
Any number
Greatest Common
Factor
a3b2+2a2b-4ab2 =
ab(a2b+2a-4b)
Two
Diff. of 2 squares
Sum of 2 cubes
Diff. of 2 cubes
a2-b2=(a+b)(a-b)
a3+b3=(a+b)(a2-ab+b2) a3b3=(a-b)(a2+ab+b2)
Three
Perfect Square
Trinomials
a2+2ab+b2=(a+b)2 a22ab+b2=(a-b)2
General Trinomials
acx2+ (ad+bc)x +bd= (ax
+b)(cx +d)
Grouping
ax+bx+ay+by=
x(a+b)+y(a+b)= (a+b)(x+y)
Four or more
2. a3 - 4a2 + 3a - 12
3. 5x2-13x + 6
4. 3xy2 – 48x
5. c3d3 + 27
6. m6 - n6
Simplify Quotients
Use long division, synthetic division, or factoring.
7.
x2 + 2x _- 3
x2 + 7x + 12
Chapter 5: Polynomials
Section 5: Roots of Real Numbers
Simplify Radicals
Examples
Finding the square root of a number and squaring a number are
_____________ operations. To find the __________ ______ of a
number n, you must find a number whose square is n.
Def of a square root: For any real numbers a and b, if a2=b, then a is a
square root of b.
Find the roots.
1. 25x4
2. -(y2 + 2)8
The inverse of raising a number to the nth power is finding the nth
root of that number.
3.
a3 = 125  555 = 125  5 is a cube root of 125
5
32 x 15y20
a4 = 81  3333 = 81  3 is a fourth root of 81
So an = b  aaa…a = b, then a is an nth root of b.
9
4.
(n times)
Def. of nth root: For any real numbers a and b, and any positive
integer n, if an = b, then a is an nth root of b.
5.
Some numbers have more than one real nth root, for example , 36
has two square roots, ______ and _______. When there is more
than one real root, the non-negative root is called the ___________
root.
Approximate radicals with a
calculator.
- 16 = -4 (opposite principal square root
3
16 = + 4 (both square roots)
 125 = -5 (principle (and only) root)
- 4 81 = -3 (opposite of principal square root)
n
n
b if b>0
n
b if b<0
even
one pos. root one
neg. root
no real roots
odd
one pos root no
neg. roots
no pos. roots
one neg.root
Chapter 5: Polynomials
x8
6. The time T in seconds that
it takes a pendulum to make a
complete swing back and
forth is given by the formula
16 = 4 (principal square root)
+
8
b= 0
T=2  L / g where L is the
length of the pendulum in
feet and g is the acceleration
due to gravity, 32 ft/sec2 .
Find the value of T for a 3ft.
pendulum in a grandfather
clock.
One real root,
0
Section 6: Radical Expressions
Simplify Radical Expressions
indices
Examples
Simplify:
radicand
Product Property of Radicals: For any real numbers a and b and any
integer n>1;
1. 16p8q7
If n is __________ and a and b are both nonnegative, then
n
ab =
n
a  n b and if n is _____, then
n
ab =
n
a n b .
2. x4 / y5
___________ Property of Radicals: For any real numbers a and b≠0,
and any integer n>1;
If all roots are defined,
n
a/b =
n
a/ nb
3.
5
5 / 4a
A radical expression is in ____________form if all of these conditions
are met:

The index n is as _______as possible

The radicand contains no ________ (other than 1) that are
nth powers of and integer or polynomial.

The radicand contains no __________.

No radicals appear in a __________________.
4. 6 3 9n 2  33 24n
5. 2 12 - 3 27 + 2 48
To eliminate radicals from a denominator or fractions from a
radicand, use “______________ the ______________” by multiplying
the numerator and denominator by a quantity so that the radicand
has an exact root.
Operations with Radicals
6.(3 5 - 2 3 )(2 +
Two radical expressions are called ________ ___________
______________ if both the indices and the radicand are alike.
Examples:
3 and 3 3 are not like expressions (different _________)
4
5 and 4 5x are not like expressions (different _________)
2 4 3a and 5 4 3a are like expressions
7. 1 - 3
5+
3
3)
Chapter 5: Polynomials
Section 7: Rational Exponents
Rational Exponents and Radicals
Squaring a number and taking the square root of a number are
____________ operations.
So (b ) = b  b
square equals b.
1/2 2
1/2
1/2
1/2+1/2
=b
We can define b1/2 =
1
1/2
= b = b and b
is a number whose
3
8 =2
And for any nonzero real number b, and any integers m and n, with
n>1, bm/n =
n
Example:
42/5 = ( 5 4 )2
Write in radical form:
1. a1/4
2. x1/5
Write using rational
exponents.
b
For any real number b and for any positive integer n, b1/n= n b except
when b<0 and n is even.
Example: 81/3 =
Examples
b m = ( n b )m
3.
3
y
4.
8
c
Evaluate:
5. 16-1/4
6. 2433/5
7. x1/5  x7/5
8. y-3/4
Simplifying Expressions
All of the properties of powers learned previously will be used. When
simplifying expressions containing rational exponents, leave the
exponent in _________ form rather than writing the expression as a
radical. To simplify, write the expression with all
_____________exponents and ____________ ___________ in the
denominator.
9.
10.
8
81 /
4
6
3
9z 2
11. (m1/2 - 1)/(m1/2 + 1)
Simplified:
-No negative exponents
-No fractional exp. In denom.
-Not a complex fraction
-Index is least number poss.
Chapter 5: Polynomials
Section 8: Radical Equations and Inequalities
Solve Radical Equations
Examples
Equations with radicals that have variables in the radicands are called
____________ __________. To solve, __________ each side of the
equation to a power equal to the index of the radical to eliminate the
radical.
When you solve a radical equation, it is very important to
____________ your solution. Sometimes you will obtain a number
that does not satisfy the original equation, called an
_______________ solution. You can use a graphing calculator to
__________ the number of solutions of an equation or to determine
whether the solution you obtained is reasonable.
1.
x 1 + 2 = 4
2.
x  15 = 3 -
x
3. 3(5n -1)1/3 – 2 = 0
Solve Radical Inequalities
A ___________ ___________ Is an inequality that has a variable in
the radicand.
4. 2 +
4x  4 < 6
Step 1: If the index of the root is _________, identify the values of
the variable for which the radicand is nonnegative.
Step 2: Solve the inequality ________________.
Step 3: Test ________ to check your solution.
Chapter 5: Polynomials
Section 9: Complex numbers
Add and Subtract Complex Numbers
Since i is defined to have the property that i2 = -1, the number i is
the principal square root of -1; that is i =  1 , i is called the ___________ _____. Numbers of the form 3i, -5i, and i 2 are
called ______ _____________ _________. They are square
roots of negative real numbers.
i =
1
i5 =
i2 = -1
i6 =
i3 = -  1
i7 =
i4 = 1
i8 =
A _____________ _________ is any number that can be written
in the form of a + bi, where a and b are real numbers and I is the
imaginary unit. a is called the real part and b is called the
imaginary part.
Two complex numbers are equal iff their real parts are equal and
their imaginary parts are equal.
Examples
1.
 18
2.
 125 x 5
3. -2i7i
4.
 10 
 15
5. i45
6. 3x2 + 48 = 0
7. Find values of x and y that
make the equation true: 2x –
3 + (y –4)i= 3 + 2i
Two complex numbers are called ________ _______________
if they are in the form of a+bi and a-bi.
To solve a division problem with complex numbers, multiply the
_____________ and __________ with the ______________or a
complex number.
Simplify:
8. (6 – 4i) + (1 + 3i)
Example:
3i  (2 + 4i)
(5 + 2i)  2i
9. (3 – 2i) – (5 – 4i)
10.In an AC circuit, the
voltage E, current I, and
impedance Z are related by the
formula E = IZ. Find the
voltage in a circuit with
current 1+3j amps and
impedance 7-5j ohms.