Algebra 2 Interactive Chalkboard
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Lesson 5-1 Monomials
Lesson 5-2 Polynomials
Lesson 5-3 Dividing Polynomials
Lesson 5-4 Factoring Polynomials
Lesson 5-5 Roots of Real Numbers
Lesson 5-6 Radical Expressions
Lesson 5-7 Rational Exponents
Lesson 5-8 Radical Equations and Inequalities
Lesson 5-9 Complex Numbers
Example 1 Simplify Expressions with Multiplication
Example 2 Simplify Expressions with Division
Example 3 Simplify Expressions with Powers
Example 4 Simplify Expressions Using
Several Properties
Example 5 Express Numbers in Scientific Notation
Example 6 Multiply Numbers in Scientific Notation
Example 7 Divide Numbers in Scientific Notation
Definition of exponents
Commutative Property
Answer:
Definition of exponents
Answer:
Subtract exponents.
Remember that a simplified
expression cannot contain
negative exponents.
1
1
Answer:
1
1
Simplify.
Answer:
Power of a power
Answer:
Answer:
Power of a power
Power of a quotient
Power of a product
Answer:
Negative exponent
Power of a quotient
Answer:
Simplify each expression.
a.
Answer:
b.
Answer:
c.
Answer:
d.
Answer:
Method 1
Raise the numerator and the denominator to
the fifth power before simplifying.
Answer:
Method 2 Simplify the fraction before raising to the fifth
power.
Answer:
Answer:
Express 4,560,000 in scientific notation.
4,560,000
Answer:
Write 1,000,000
as a power of 10.
Express 0.000092 in scientific notation.
Answer:
Use a negative exponent.
Express each number in scientific notation.
a. 52,000
Answer:
b. 0.00012
Answer:
Express the result in
scientific notation.
Associative and
Commutative Properties
Answer:
Express the result in
scientific notation.
Associative and
Commutative Properties
Answer:
Evaluate. Express the result in scientific notation.
a.
Answer:
b.
Answer:
Biology There are about
red blood cells in one
milliliter of blood. A certain blood sample contains
red blood cells. About how many milliliters
of blood are in the sample?
Divide the number of red blood cells in the sample by the
number of red blood cells in 1 milliliter of blood.
 number of red blood cells in sample
 number of red blood cells in 1 milliliter
Answer: There are about 1.66 milliliters of blood
in the sample.
Biology A petri dish started
with
germs in it.
A half hour later, there are
How many times as
great is the amount a half
hour later?
Answer:
Example 1 Degree of a Polynomial
Example 2 Subtract and Simplify
Example 3 Multiply and Simplify
Example 4 Multiply Two Binomials
Example 5 Multiply Polynomials
Determine whether
is a polynomial. If it
is a polynomial, state the degree of the polynomial.
Answer: This expression is not a polynomial
because
is not a monomial.
Determine whether
is a polynomial.
If it is a polynomial, state the degree of the polynomial.
Answer: This expression is a polynomial because each
term is a monomial. The degree of the first term
is 5 and the degree of the second term is 2 + 7
or 9. The degree of the polynomial is 9.
Determine whether each expression is a polynomial. If
it is a polynomial, state the degree of the polynomial.
a.
Answer: yes, 5
b.
Answer: no
Simplify
Distribute the –1.
Group like terms.
Answer:
Combine like terms.
Simplify
Answer:
Distributive Property
Answer:
Multiply the monomials.
Answer:
+
+
+
First terms Outer terms Inner terms Last terms
Answer:
Multiply monomials
and add like terms.
Answer:
Distributive Property
Distributive Property
Multiply monomials.
Answer:
Combine like terms.
Answer:
Example 1 Divide a Polynomial by a Monomial
Example 2 Division Algorithm
Example 3 Quotient with Remainder
Example 4 Synthetic Division
Example 5 Divisor with First Coefficient Other than 1
Sum of quotients
Divide.
Answer:
Answer:
Use long division to find
Answer:
Use long division to find
Answer: x + 2
Multiple-Choice Test Item
Which expression is equal to
A
B
C
D
Read the Test Item
Since the second factor has an exponent of –1,
this is a division problem.
Solve the Test Item
The quotient is –a + 3 and the remainder is –3.
Therefore,
Answer: D
Multiple-Choice Test Item
Which expression is equal to
A
B
C
D
Answer: B
Use synthetic division to find
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.
x3 – 4x2 + 6x – 4
 


1 –4
6 –4
Step 2 Write the constant r of the
divisor x – r to the left. In this
case, r = 2. Bring the first
coefficient, 1, down as shown.
1 –4
1
6 –4
Step 3 Multiply the first coefficient by
. Write the product
under the second coefficient.
Then add the product and the
second coefficient.
1 –4
2
1 –2
6 –4
Step 4 Multiply the sum, –2, by
Write the
product under the next
coefficient and add:
1 –4 6 – 4
2 –4
1 –2 2
Step 5 Multiply the sum, 2, by
Write the product
under the next coefficient and
add:
The remainder is 0.
1 –4 6
2 –4
1 –2 2
–4
4
0
The numbers along the bottom are the coefficients of the
quotient. Start with the power of x that is one less than the
degree of the dividend.
Answer: The quotient is
Use synthetic division to find
Answer: x + 7
Use synthetic division to find
Use division to rewrite the divisor so it has a first
coefficient of 1.
Divide numerator and
denominator by 2.
Simplify the numerator
and denominator.
Since the numerator does not have a y3 term, use a
coefficient of 0 for y3.
The result is
Now simplify the fraction.
2
1 –2
0
2
Rewrite as a
division expression.
Multiply by
the reciprocal.
Multiply.
Answer: The solution is
Use synthetic division to find
Answer:
Example 1 GCF
Example 2 Grouping
Example 3 Two or Three Terms
Example 4 Quotient of Two Trinomials
Factor
The GCF
is 5ab.
Answer:
Distributive
Property
Factor
Answer:
Factor
Group to find the GCF.
Factor the GCF of
each binomial.
Answer:
Distributive Property
Factor
Answer:
Factor
To find the coefficient of the y terms, you must find two
numbers whose product is 3(–5) or –15 and whose sum
is –2. The two coefficients must be 3 and –5 since
and
.
Rewrite the expression using –5y and 3y in place
of –2y and factor by grouping.
Substitute –5y + 3y for –2y.
Associative Property
Factor out the GCF of each group.
Answer:
Distributive Property
Factor
Factor out the GCF.
Answer:
p2 – 9 is the difference
of two squares.
Factor
This is the sum of two cubes.
Sum of two cubes formula
with
and
Answer:
Simplify.
Factor
This polynomial could be considered the difference of
two squares or the difference of two cubes. The
difference of two squares should always be done before
the difference of two cubes.
Difference of two squares
Answer:
Sum and difference
of two cubes
Factor each polynomial.
a.
Answer:
b.
Answer:
c.
Answer:
d.
Answer:
Simplify
Factor the numerator
and the denominator.
Divide. Assume
a  –5, –2.
Answer: Therefore,
Simplify
Answer:
Example 1 Find Roots
Example 2 Simplify Using Absolute Value
Example 3 Approximate a Square Root
Simplify
Answer: The square roots of 16x6 are  4x3.
Simplify
Answer: The opposite of the principal square root of
Simplify
Answer: The principal fifth root is 3a2b3.
Simplify
Answer: n is even and b is negative.
Thus,
is not a real number.
Simplify.
a.
Answer:  3x4
b.
Answer:
c.
Answer: 2xy2
d.
Answer: not a real number
Simplify
Note that t is a sixth root of t6. The index is even, so the
principal root is nonnegative. Since t could be negative,
you must take the absolute value of t to identify the
principal root.
Answer:
Simplify
Since the index is odd, you do not need absolute value.
Answer:
Simplify.
a.
Answer:
b.
Answer:
Physics The time T in seconds that it takes a
pendulum to make a complete swing back and forth is
given by the formula
where L is the length
of the pendulum in feet and g is the acceleration due
to gravity, 32 feet per second squared. Find the value
of T for a 1.5-foot-long pendulum.
Explore You are given the values of L and g and
must find the value of T. Since the units on
g are feet per second squared, the units on
the time T should be seconds.
Plan
Substitute the values for L and g into the
formula. Use a calculator to evaluate.
Solve
Original formula
Use a calculator.
Answer: It takes the pendulum about 1.36 seconds
to make a complete swing.
Examine The closest square to
and
 is approximately 3, so the answer should
be close to
The answer is reasonable.
Physics The time T in seconds that it takes a
pendulum to make a complete swing back and forth is
given by the formula
where L is the length
of the pendulum in feet and g is the acceleration due
to gravity, 32 feet per second squared. Find the value
of T for a 2-foot-long pendulum.
Answer: about 1.57 seconds
Example 1 Square Root of a Product
Example 2 Simplify Quotients
Example 3 Multiply Radicals
Example 4 Add and Subtract Radicals
Example 5 Multiply Radicals
Example 6 Use a Conjugate to Rationalize
a Denominator
Simplify
Factor into squares
where possible.
Product Property
of Radicals
Answer:
Simplify.
Simplify
Answer:
Simplify
Quotient Property
Factor into squares.
Product Property
Rationalize the denominator.
Answer:
Simplify
Quotient Property
Rationalize the denominator.
Product Property
Multiply.
Answer:
Simplify each expression.
a.
Answer:
b.
Answer:
Simplify
Product Property
of Radicals
Factor into cubes.
Product Property
of Radicals
Answer:
Multiply.
Simplify
Answer: 24a
Simplify
Factor using squares.
Product Property
Multiply.
Combine like radicals.
Answer:
Simplify
Answer:
Simplify
F
O
I
L
Product Property
Answer:
Simplify
FOIL
Multiply.
Answer:
Subtract.
Simplify each expression.
a.
Answer:
b.
Answer: 41
Simplify
Multiply by
since
is the conjugate
of
FOIL
Difference of
squares
Multiply.
Answer:
Combine like terms.
Simplify
Answer:
Example 1 Radical Form
Example 2 Exponential Form
Example 3 Evaluate Expressions with
Rational Exponents
Example 4 Rational Exponent with Numerator Other
Than 1
Example 5 Simplify Expressions with
Rational Exponents
Example 6 Simplify Radical Expressions
Write
Answer:
in radical form.
Definition of
Write
Answer:
in radical form.
Definition of
Write each expression in radical form.
a.
Answer:
b.
Answer:
Write
Answer:
using rational exponents.
Definition of
Write
Answer:
using rational exponents.
Definition of
Write each radical using rational exponents.
a.
Answer:
b.
Answer:
Evaluate
Method 1
Answer:
Simplify.
Method 2
Power of a Power
Multiply exponents.
Answer:
Evaluate
.
Method 1
Factor.
Power of a Power
Expand the square.
Find the fifth root.
Answer: The root is 4.
Method 2
Power of a Power
Multiply exponents.
Answer: The root is 4.
Evaluate each expression.
a.
Answer:
b.
Answer: 8
Weight Lifting The formula
can be used to estimate the maximum total mass that
a weight lifter of mass B kilograms can lift in two lifts,
the snatch and the clean and jerk, combined.
According to the formula, what is the maximum that
U.S. Weightlifter Oscar Chaplin III can lift if he weighs
77 kilograms?
Original formula
Use a calculator.
Answer: The formula predicts that he can
lift at most 372 kg.
Weight Lifting The formula
can be used to estimate the maximum total mass that
a weight lifter of mass B kilograms can lift in two lifts,
the snatch and the clean and jerk, combined.
Oscar Chaplin’s total in the 2000 Olympics was
355 kg. Compare this to the value predicted by the
formula.
Answer: The formula prediction is somewhat
higher than his actual total.
Weight Lifting Use the formula
where M is the maximum total mass that a weight
lifter of mass B kilograms can lift.
a. According to the formula, what is the maximum that a
weight lifter can lift if he weighs 80 kilograms?
Answer: 380 kg
b. If he actually lifted 379 kg, compare this to the value
predicted by the formula.
Answer: The formula prediction is slightly higher than his
actual total.
Simplify
.
Multiply powers.
Answer:
Add exponents.
Simplify
.
Multiply by
Answer:
Simplify each expression.
a.
Answer:
b.
Answer:
Simplify
.
Rational exponents
Power of a Power
Quotient of Powers
Answer:
Simplify.
Simplify
.
Rational exponents
Power of a Power
Multiply.
Answer:
Simplify.
Simplify
.
is the conjugate
of
Answer:
Multiply.
Simplify each expression.
a.
Answer: 1
b.
Answer:
c.
Answer:
Example 1 Solve a Radical Equation
Example 2 Extraneous Solution
Example 3 Cube Root Equation
Example 4 Radical Inequality
Solve
Original equation
Add 1 to each side to
isolate the radical.
Square each side to
eliminate the radical.
Find the squares.
Add 2 to each side.
Check
Original equation
Replace y with 38.
Simplify.
Answer: The solution checks. The solution is 38.
Solve
Answer: 67
Solve
Original equation
Square each side.
Find the squares.
Isolate the radical.
Divide each side by –4.
Square each side.
Evaluate the squares.
Check
Original equation
Replace x with 16.
Simplify.
Evaluate the square roots.
Answer: The solution does not check, so there is
no real solution.
Solve
Answer: no real solution
.
Solve
In order to remove the
power, or cube root, you must
first isolate it and then raise each side of the equation to
the third power.
Original equation
Subtract 5 from each side.
Cube each side.
Evaluate the cubes.
Subtract 1 from each side.
Divide each side by 3.
Check
Original equation
Replace y with –42.
Simplify.
The cube root of –125 is –5.
Add.
Answer: The solution is –42.
Solve
Answer: 13
Solve
Since the radicand of a square root must be greater
than or equal to zero, first solve
to identify
the values of x for which the left side of the inequality
is defined.
Now solve
.
Original inequality
Isolate the radical.
Eliminate the radical.
Add 6 to each side.
Divide each side by 3.
Answer: The solution is
Check
Test some x values to confirm the solution. Let
Use three test values: one less than
2, one between 2 and 5, and one greater than 5.
Since
the
Since
is not a real Since
number, the inequality inequality is satisfied. the inequality is
not satisfied.
is not satisfied.
Only the values in the interval
satisfy the inequality.
Solve
Answer:
Example 1 Square Roots of Negative Numbers
Example 2 Multiply Pure Imaginary Numbers
Example 3 Simplify a Power of i
Example 4 Equation with Imaginary Solutions
Example 5 Equate Complex Numbers
Example 6 Add and Subtract Complex Numbers
Example 7 Multiply Complex Numbers
Example 8 Divide Complex Numbers
Simplify
Answer:
.
Simplify
Answer:
.
Simplify.
a.
Answer:
b.
Answer:
Simplify
Answer:
.
=6
Simplify
Answer:
.
Simplify.
a.
Answer: –15
b.
Answer:
Simplify
Multiplying powers
Power of a Power
Answer:
Simplify
Answer: i
.
Solve
Original equation
Subtract 20 from
each side.
Divide each side by 5.
Take the square
root of each side.
Answer:
Solve
Answer:
Find the values of x and y that make the equation
true.
Set the real parts equal to each other and the imaginary
parts equal to each other.
Real parts
Divide each side by 2.
Imaginary parts
Answer:
Find the values of x and y that make the equation
true.
Answer:
Simplify
.
Commutative
and Associative
Properties
Answer:
Simplify
.
Commutative
and Associative
Properties
Answer:
Simplify.
a.
Answer:
b.
Answer:
Electricity In an AC circuit, the voltage E, current I,
and impedance Z are related by the formula
Find the voltage in a circuit with current 1 + 4 j amps
and impedance 3 – 6 j ohms.
Electricity formula
FOIL
Multiply.
Add.
Answer: The voltage is
volts.
Electricity 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 – 3 j amps
and impedance 3 + 2 j ohms.
Answer: 9 – 7 j
Simplify
.
and
are conjugates.
Multiply.
Answer:
Standard form
Simplify
.
Multiply by
Multiply.
Answer:
Standard form
Simplify.
a.
Answer:
b.
Answer:
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