fundamental skills of algebra slide 3 / 224
TRANSCRIPT
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Algebra II
Fundamental Skills of
Algebra
www.njctl.org
2013-09-11
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Table of Contents
Solving Equations and Inequalities
Factoring
Exponents
Radicals
click on the topic to go to that section
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Solving Equations and Inequalities
Return to Table of Contents
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Goals and ObjectivesStudents will be able to solve a wide variety of equations and inequalities.
Solving Equations and Inequalities
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Solving Equations and Inequalities
Why do we need this?Life throws at us many different types of
problems. Being able to solve these problems makes us successful.
Developing logic skills that we can apply in different situations helps us solve
those problems with ease.
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Solving Equations and Inequalities
Steps for solving equations:
1) Remove any parentheses.2) Collect like terms.3) Move all variables to one side.4) Move all constants to one side.5) Remove any coefficient.
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Solving Equations and Inequalities
Remember...
· There are typically two "sides" to an equation. They are separated by the equals sign.
· An equation is like a balance. What you do to one side, must be done to the other side!
· Use opposite operations (+/-, x/÷) when moving variables and constants across the equals sign.
· "Collecting like terms" is just organizing and simplifying each side of the equation. Do not use opposites when collecting like terms.
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1 Try on your own. Use the rules from your notes to help you.
Solving Equations and Inequalities
-3(2x + 4) - 8 = 2(x - 5) - 2
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Solving Equations and Inequalities
The nice part about how equations work is that you can always check your answer. Plug in your result to the original equation even if your first answer was incorrect. What happens?
-3(2x + 4) - 8 = 2(x - 5) - 2 -3(2x + 4) - 8 = 2(x - 5) - 2Plug in incorrect answer: Plug in correct answer:
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Solving Equations and Inequalities
More examples:
3m - 4 + 2m = 5 + 4m
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Solving Equations and Inequalities
More examples:
-2(a + 3) - 2 = 3a - 15 - 8
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Solving Equations and Inequalities
and a few more... Will all of our answers be nice, whole numbers?
-4y - 5 + 3y - 6 = 4 + 2y + 7
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Solving Equations and Inequalities
and a few more... Will all of our answers be nice, whole numbers?
3(x - 4) + 5 = 1 + 7(x - 2)
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2 Solve for x: 2(x - 4) - 3(x + 5) = 2x - 8
Solving Equations and Inequalities
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3 Find m: 5m - 4 - 2m = 4m - 12 + 6Solving Equations and Inequalities
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4 Solve the equation: -6(y + 2) = -7(y - 3)
Solving Equations and Inequalities
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5 Find x: 5 - 2x - 4 - 4x = -3 - 5x + 2x - 5
Solving Equations and Inequalities
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6 Solve for x: -4 - (x - 5) = 2 - (3x - 8)Solving Equations and Inequalities
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Try:
Solving Equations and Inequalities
-2 - (m - 3) = -(m - 4) - 2
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Solving Equations and Inequalities
Sometimes, the variable cancels out and you are left with a numerical statement. If that happens, you will have one of two possible answers.
True Statement = infinitely many solutionsFalse Statement = no solution
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For example:
-p + 3 = -p + 6
results in 3 ≠ 6
This is a false statement and
there is no solution.
b - 8 = b - 8
results in -8 = -8
This is a true statement and
there are infinitely many
solutions.
Solving Equations and Inequalities
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7 What is the solution to the equation: 5 - 2(x + 3) = -2x + 4
A -4B 0C No solutionD Infinitely many solutions
Solving Equations and Inequalities
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8 Find x: -2(x - 3) + 4 = -x + 10
A 10B 0C No solutionD Infinitely many solutions
Solving Equations and Inequalities
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9 Solve: m - 6 - 3 - 2m = m - 4 - m - 5 - m
A 9B 0C No solutionD Infinitely many solutions
Solving Equations and Inequalities
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10 Find y: -4y + 8 - y = y - 6 - 8
A 11/3B 0C No solutionD Infinitely many solutions
Solving Equations and Inequalities
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11 Find the answer to the equation: -2(x - 4) + 5(x + 3) = 3x - 12 + 2
A 4B 0C No solutionD Infinitely many solutions
Solving Equations and Inequalities
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Solving Equations and Inequalities
Now for fractions...
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Try:
Solving Equations and Inequalities
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12 Solve:
Solving Equations and Inequalities
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13 Find x:
Solving Equations and Inequalities
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14 Solve the equation:
Solving Equations and Inequalities
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15 Find m:
Solving Equations and Inequalities
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16 Solve the equation:
Solving Equations and Inequalities
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Solving Equations and Inequalities
Solving formulas for specific variables incorporates all of these rules. Try...
4pc = 2t - 9dm solve for d
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Solving Equations and Inequalities
Try...this one is a bit tougher...
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17 Solve the following formula for C.
A
B
C
D
E
Solving Equations and Inequalities
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18 Solve for h:
A
B
C
D
E
Solving Equations and Inequalities
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19 Solve for h:
A
B
C
D
E
Solving Equations and Inequalities
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20 Solve for w:
A
B
C
D
E
Solving Equations and Inequalities
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21 Solve for m:
A
B
C
D
E
Solving Equations and Inequalities
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22 Solve for x:
A
B
C
D
E
Solving Equations and Inequalities
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Solving Equations and Inequalities
Remember inequalities?
≥, ≤ >, <[ , ] ( , )
All symbols indicate solutions will include referenced points.
All symbols indicate solutions will not include referenced points.
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Solving Equations and Inequalities
You solve inequalities the same way that you solve equations. The only difference is that you flip an inequality symbol if you multiply or divide by a negative number.
Solve: 3m + 4 - 5m < 3m + 6
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Solutions to equations are single points. Solutions to inequalities are regions of points.
Solving Equations and Inequalities
Draw a graph to represent the solution to the last problem: m > -2/5
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Solving Equations and Inequalities
Try solving and graphing the inequality...
2x - 6 + 3 ≥ 5x - 4
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One more, solve and graph the inequality...
Solving Equations and Inequalities
3(x - 2) - 4(x + 3) ≤ -2(x - 4)
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23 Solve for x: 5(x - 3) + 2 -(2x - 4)
A
B
CDE
Solving Equations and Inequalities
x ≤ 17/7
x ≥ 17/2
x ≤ 21/5x ≤ 14/5
x ≥ 21/5
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24 Solve the following inequality: 6y - 2 - 2y > 3y + 5
A y > 7/9B y < 7/9C y > 7/5D y > 3/5E y > 7
Solving Equations and Inequalities
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25 Find values for p: 2(p + 3) - 5 4(p + 4)
A
B
C
D
E Answer not listed.
Solving Equations and Inequalities
p ≥ -15/2
p ≤ -15/2
p ≥ -5/2
p ≤ -5/2
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26 Which of the following numbers would be a solution to the following inequality? 3p + 6 - 5p > 10p - 4 - 8
A 0B 5/2C 5D 15/4E 9
Solving Equations and Inequalities
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27 Solve for n: 4 - 3(n - 3) 2(n + 4) - 2
A
B
C
D
E Answer not listed.
n ≤ -11/5
n ≥ -11/5n ≥ 19/5
n ≤ 19/5
Solving Equations and Inequalities
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Factoring
Return to Table of Contents
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Factoring
Goals and ObjectivesStudents will be able to factor complex expressions and solve equations using factoring.
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Factoring
Why do we need this?The more we can simplify a problem, the
easier it is to solve. Factoring allows us to break up expressions into smaller parts
and, much of the time, simplify our strategies to solve them.
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Factoring
Multiply the following:
3x(2x2 + 5) (x + 3)(x - 9) (2x - 1)(3x - 4)
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Factoring
Factoring is undoing what you just did. It breaks up expressions into parts and pieces.
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Factoring
Factor out the GCF (greatest common factor).
8x + 32y 5a2b - 10ab 12a3b2 + 4ab2
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Factoring
You can easily check your answer by distributing the GCF back to all of your terms.
Factor then check your answer:
3x2y2 + 12xy2 + 6y2 6m4n3 + 18m3n2 - 36m2n
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28 What is the GCF of the following expression? 24a3b3c - 6ab3c
A 6a3b3c B 6abc C ab3c D 6ab3c E a3b3c
Factoring
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29 Factor:
A 3m5n2(4 - n - 5m)
B m5n2(12 - 3n - 15m)
C 3m6n3(4 - 5n)
D 3m4n2(4mn - 5m2)
E 3m4n2(4 - mn - 5m2)
Factoring
12m4n2 - 3m5n3 - 15m6n2
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30 Factor out the GCF:
A x2y2z3(x2 - xy + y2)
B xyz(x2 - xy + y2)
C x3y2z3(x - y - y2)
D x2y3z3(x2 - x - y)
E x2y2z3(x - xy + y2)
x4y2z3 - x3y3z3 + x2y4z3
Factoring
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31 Factor the following:
A -4x3y2(2x + 3y)
B -4x3y2(2x - 3y)
C -8x3y2(x + 4y)
D -8x3y2(x - 4y)
E -8x4y2(1 - 4y)
-8x4y2 - 12x3y3
Factoring
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32 Factor out the GCF: 15m3n - 25m2 - 15mn3
A 15m(mn - 10m - n3)
B 5m(3m2n - 5m - 3n3)
C 5mn(3m2 - 5m - 3n2)
D 5mn(3m2 - 5m - 3n)
E 15mn(mn - 10m - n3)
Factoring
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33 Factor:
A m3n3(2m - 2)
B -2mn(m3n + m)
C -2mn(m3n - m)
D -2m3n3(m2 + 1)
E -2m3n3(m2 - 1)
Factoring
-2m4n3 - 2m3n3
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34 Factor out the GCF:
A 14p3q7(1p - 14)
B 14p2q6(pq - 14)
C 14p2q6(pq - 2)
D p3q6(14q - 28)
E 14pq(p2q6 - 2pq5)
Factoring
14p3q7 - 28p2q6
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Factoring
Factoring quadratics of the form x2 + bx + c.
Try: x2 - x - 6
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Factoring
Here are some tips to help you factor:
Read the expression backwards:
x2 - x - 6
"factors of 6 that subtract to -1"
x2 + 6x + 8
"factors of 8 that add up to 6"
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Factoring
The signs in front of c will also help you factor:
x2 + bx + c ⇒ (x + m)(x + n)
x2 - bx + c ⇒ (x - m)(x - n)
x2 + bx - c
x2 - bx - c
If the sign in front of c is +: If the sign in front of c is -:
(x - m)(x + n)⇒
⇒
same signs, both +
same signs, both -
different signs
different signs
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Factoring
Here is the last tip...
Use a factor tree to help you find the factors of c. If none of the pairs add or subtract to the middle term, the quadratic is not factorable.
Factor trees: 12
>
1 122 63 4
24
>
1 242 123 84 6
48
>
1 482 243 164 126 8
72
>
1 722 363 244 186 128 9
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Factoring
x2 + 6x + 8
Read it backwards: "factors of 8 that add up to 6."
Use the pattern from c: "same signs, both +."
Now factor it!
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Factoring
Use the tips to help you factor the following quadratics.
x2 - 5x - 24 a2 - 13a + 30 m2 + 4m - 35
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Factoring
Try...x2 - 8xy + 16y2
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35 Factor the trinomial:
A (x - 7y)(x + 6y)
B (x + 7y)(x - 6y)
C (x - 42y)(x + y)
D (x + 42y)(x - y)
E Solution not shown
Factoring
x2 - xy - 42y2
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36 Factor:
A (x - 12)(x + 2)
B (x + 12)(x - 2)
C (x - 6)(x +4)
D (x - 6)(x - 4)
E Solution not shown
Factoring
x2 - 10x + 24
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37 Factor the quadratic:
A (x + 5)(x + 1)
B (x - 4)(x - 1)
C (x + 2)(x + 3)
D (x + 1)(x - 4)
E Solution not shown
x2 + 5x + 4
Factoring
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38 Factor:
A (x - 12)(x - 6)
B (x - 6)(x - 3)
C (x - 6)(x + 3)
D (x - 9)(x + 2)
E Solution not shown
x2 - 3x - 18
Factoring
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39 Factor:
A (x - 5)(x - 5)
B (x - 5)(x + 5)
C (x + 15)(x + 10)
D (x - 15)(x - 10)
E Solution not shown
Factoring
x2 + 10x + 25
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FactoringFactoring quadratics of the form ax2 + bx + c.
3x2 + 17x + 10
With just a few extra steps, you can factor quadratics with a leading coefficient (a) just like the previous ones.
Mulitiply a and c ⇒ x2 + 17x + 30Factor the result ⇒ (x + 15)(x + 2)
Divide the numbers by a ⇒ (x + 15)(x + 2) 3 3
Reduce any fractions ⇒ (x + 5)(x + 2)3
Move any remaining denominators ⇒ (x + 5)(3x + 2)
The answer is (x + 5)(3x + 2)
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Factoring
Try this one. Make sure you reduce the fractions where possible.
10x2 - 31x + 15
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Factoring
Here is another one to practice on...
3x2 + 5x - 2
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Factoring
Try... 4x2 + 4x +1
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40 Factor:
A (6x - 5)(x - 6) B (6x - 1)(x - 6) C (3x - 2)(2x + 3) D (3x + 2)(2x - 3) E Solution not shown
Factoring
6x2 - 5x - 6
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41 Factor the following:
A (2x - 1)(5x + 3) B (2x + 1)(5x + 3) C (10x - 1)(x + 3) D (10x - 1)(x - 3) E Solution not shown
10x2 - 11x + 3
Factoring
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42 Which is a factor of ?
A (4x + 5)B (2x + 3)C (12x + 5)D (3x + 5)E (4x + 2)
12x2 + 23x + 10
Factoring
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43 Which is a factor of the quadratic?
A (x - 7y)B (2x - 8y)C (2x - 6y)D (x - 8y)E (2x - y)
Factoring
2x2 - 23xy + 56y2
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44 Which of the following is a factor of ?
A (2x - 3y)B (5x - 3y)C (3x - 2y)D (3x + 5y)E (2x - 5y)
6x2 - 19xy + 15y2
Factoring
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45 Factor:
A (x - 4y)(4x + 2y)B (x - 8y)(4x + y)C (2x + 4y)(2x + 2y)D (4x + y)(x + 8y)E Solution not shown
Factoring
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Factoring
Multiply:
(2x + 3)(2x - 3) (x + 3)(x2 - 3x + 9)
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To factor the difference of squares, the difference of cubes and the sum of cubes, use the following formulas:
Factoring
a2 - b2 ⇒ (a - b)(a + b)
a3 - b3 ⇒ (a - b)(a2 + ab + b2)
a3 + b3 ⇒ (a + b)(a2 - ab + b2)
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Try...
Factoring
4p2 - q2 16m2 - 1 64p3 + y3
a2 - b2 ⇒ (a - b)(a + b)a3 - b3 ⇒ (a - b)(a2 + ab + b2)a3 + b3 ⇒ (a + b)(a2 - ab + b2)
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Factoring
a2 - b2 ⇒ (a - b)(a + b)a3 - b3 ⇒ (a - b)(a2 + ab + b2)a3 + b3 ⇒ (a + b)(a2 - ab + b2)Factor...
25x2 - 81y2 x3y3 + 1 8m3 - 125n3
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46 Factor:
A (11m - 10n)(11m + 10m) B (121m - n)(m + 100n) C (11m - n)(11m + 100n) D Not factorable E Solution not shown
121m2 + 100n2
Factoring
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47 Factor:
A (3m - 8np)(3m + 8np) B (3m + 4np)(3m2 - 12mnp + 4n2p2) C (3m + 4np)(9m2 - 12mnp + 16n2p2) D Not factorable E Solution not shown
Factoring
27m3 + 64n3p3
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48 Factor:
A (a - 5b)(a + 5b) B (a - 5b)(a2 + 5ab + 5b2) C (a + 5b)(a2 - 5ab + 25b2) D Not FactorableE Solution not shown
Factoring
a3 - 125b3
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49 Factor:
A (m + n)(m - n) B (m + n)(m + n) C (m - n)(m - n) D Not factorableE Solution not shown
Factoring
m2 - n2
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50 Factor:
A (d - 1)(d + 1) B (d - 1)(d2 + d + 1) C (d + 1)(d2 - d + 1) D Not factorableE Solution not shown
Factoring
d3 - 1
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51 Factor:
A (3m + 2n)(3m - 2n) B (3m + 2n)(9m2 - 6mn + 4n2) C (3m - 2n)(9m2 + 6mn + 4n2) D Not factorableE Solution not shown
Factoring
27m3 + 4n3
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52 Factor:
A (16x - 2y)(16x + 2y) B (6x - 2y)(6x2 + 12xy + 2y2) C (6x - 2y)(36x2 + 12xy + 4y2) D Not factorableE Solution not shown
Factoring
216x3 - 8y3
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Factoring
Factoring by Grouping.
What happens when there are 4 terms?
4ap - 4a + 3xp - 3x
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Factoring
Try... xy + 4x - 3y - 12
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Factoring
Two more...pq + 4p + 3q + 12 mn - pm - qn + qp
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53 Factor by grouping:
A (r + 4)(s - 7) B (r - 7)(s + 4) C (rs - 7)(rs + 4) D Not factorableE Solution not shown
Factoring
rs - 7r + 4s - 28
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54 Factor:
A (n - 3)(m + 4n) B (n - 3)(m - 4n) C (n + 4)(m - n) D Not factorableE Solution not shown
Factoring
mn + 3m - 4n2 - 12n
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55 Factor:
A (3k - 2)(g + 6) B (3g + 2)(k - 6) C (3k - 2)(g + 6) D Not factorableE Solution not shown
Factoring
3gk - 18g + 2k - 12
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56 Factor by grouping:
A (m - 4)(2p - 7) B (m + 7)(2p + 4) C (m - 4)(2p + 7) D Not factorableE Solution not shown
Factoring
2mp - 8p - 7m + 28
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57 Factor by grouping
A (x + 4)(2y + 3) B (x + 4)(2y - 3) C (x - 3)(2y + 4) D Not factorableE Solution not shown
2xy + 3x + 8y - 12
Factoring
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58 Factor:
A (3m - 5)(p + 2n) B (3m + 5)(p - 2n) C (3m - 2n)(p + 5) D Not factorableE Solution not shown
Factoring
3mp + 15m - 2np - 10n
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Now, let's combine all of the situations. In any factoring problem, factor out the GCF first.
Factoring
Factor these completely...
2x3 - 22x2 + 48x 3m3n + 3m2n - 18mn
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Factoring
Factor completely...
4x3 - 32y3 54a4 + 2ab3
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59 Factor completely:
A -3mn(2m - 1)(m - 3) B 3mn(2m + 1)(m - 3) C -3n(2m - 1)(m2 + 3m + 9) D Not factorableE Solution not shown
Factoring
-6m3n + 21m2n - 9mn
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60 Factor completely:
A -3(2p + 5)(4p3 - 10p + 25) B -3p(16p2 + 25) C -3p(4p - 5)(4p - 5) D Not factorableE Solution not shown
Factoring
-48p3 + 75p
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61 Factor completely:
A (4p2 - 3)(m - 4) B 4p2(pm - 4)(pm - 3) C 4p2m(m - 4)(p + 3) D Not factorableE Solution not shown
Factoring
4p3m - 12p3 - 16mp2 + 48p2
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62 Factor completely:
A 2xy(9x - 1) B 2xy(3x - 1)(3x + 1) C 2y(9x2 - x) D Not factorableE Solution not shown
Factoring
18x3y - 2xy
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63 Factor completely:
A -8ab(a2 + 4) B -4ab(2a2 + a + 3)C -4ab(2a + 3)(a - 1)D Not factorableE Solution not shown
Factoring
-8a3b - 4a2b + 12ab
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64 Factor completely:
A 2b(4b - 1)(3b + 2) B 4b(6b2 + 6b - 1) C 2b(4b + 1)(3b - 2) D Not factorableE Solution not shown
24b3 + 10b2 - 4b
Factoring
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Factoring is often use to solve equations that are in polynomial form.
Factoring
Steps: 1) Move all terms to one side of the equation. (the other side becomes zero) 2) Factor the resulting polynomial. 3) Set each factor equal to zero. 4) Solve each equation. 5) Write the answers clearly.
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Factoring
Solve the equation by factoring:
x2 = 9x - 18
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One more...
Factoring
6x3 + 10x2 = 4x
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Another example...
Factoring
6m2 = 9m - 24m3
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65 Which of the following are solutions to the equation?
A 0B -1C -3/2D 3/2E -4F 4G -4/3H 4/3
Factoring
16x3 - 36x = 0
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66 Find all of the solutions to:
A 0B -1C 1D -3E 3F -4G 4H 12
Factoring
-3m3 + 3m2 = -36m
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67 Solve for p:
A 0B -1/2C 1/2D -2/5E 2/5F -4/3G -5/2H 5/2
Factoring
29p2 + 10p = -10p3
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68 Find the values for n:
A 0B -1/2C 1/2D -1/3E -2/3F 2/3G -4H 4
Factoring
18n4 + 48n2 = 84n3
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69 Find x:
A 0B -1/4C 1/4D -1/3E 1/3F -1/2G 1/2H 3
Factoring
6x4 = 5x3 - x2
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70 Solve by factoring:
A 0B -1/4C 1/4D -1/2E 1/2F 1G -2H 2
Factoring
p2 + p = 2p3
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Exponents
Return to Table of Contents
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Exponents
Goals and ObjectivesStudents will be able to simplify complex expressions containing exponents.
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Exponents
Why do we need this?Exponents allow us to condense bigger
expressions into smaller ones. Combining all properties of powers
together, we can easily take a complicated expression and make it
simpler.
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Exponents
Rules for working with exponents:
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Exponents
Multiplying powers of the same base:
(x4y3)(x3y)
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Exponents
(-3a3b2)(2a4b3)
Simplify:
(-4p2q4n)(3p3q3n)
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Work out:
Exponents
xy3 x5y4. (3x2y3)(2x3y)
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71 Simplify:
A m4n3p2 B m5n4p3 C mnp9 D Solution not shown
(m4np)(mn3p2)
Exponents
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72 Simplify:
A x4y5 B 7x3y5 C -12x3y4 D Solution not shown
Exponents
(-3x3y)(4xy4)
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73 Work out:
A 6p2q4 B 6p4q7 C 8p4q12 D Solution not shown
Exponents
2p2q3 4p2q4.
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74 Simplify:
A 50m6q8 B 15m6q8 C 50m8q15 D Solution not shown
Exponents
.5m2q3 10m4q5
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75 Simplify:
A a4b11 B -36a5b11 C -36a4b30 D Solution not shown
(-6a4b5)(6ab6)
Exponents
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Exponents
Dividing powers with the same base:
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Exponents
Simplify:
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Exponents
Try...
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77 Simplify:
A
B
C
D Solution not shown
Exponents
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79 Divide:
A
B
C
D Solution not shown
Exponents
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Exponents
Power to a power:
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Exponents
Simplify:
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Try:
Exponents
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81 Work out:
A
B
C
D Solution not shown
Exponents
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82 Work out:
A
B
C
D Solution not shown
Exponents
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83 Simplify:
A
B
C
D Solution not shown
Exponents
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84 Simplify:
A
B
C
D Solution not shown
Exponents
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85 Simplify:
A
B
C
D Solution not shown
Exponents
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Negative and zero exponents:Exponents
Why is this? Work out the following:
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Exponents
Sometimes it is more appropriate to leave answers with positive exponents, and other times, it is better to leave answers without
fractions. You need to be able to translate expressions into either form.
Write with positive exponents: Write without a fraction:
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Exponents
Simplify and write the answer in both forms.
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Exponents
Simplify and write the answer in both forms.
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Exponents
Simplify:
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Exponents
Write the answer with positive exponents.
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87 Simplify. The answer may be in either form.
A
B
C
D Solution not shown
Exponents
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88 Write with positive exponents:
A
B
C
D Solution not shown
Exponents
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89 Simplify and write with positive exponents:
A
B
C
D Solution not shown
Exponents
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90 Simplify. Write the answer with positive exponents.
A
B
C
D Solution not shown
Exponents
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91 Simplify. Write the answer without a fraction.
A
B
C
D Solution not shown
Exponents
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CombinationsExponents
Usually, there are multiple rules needed to simplify problems with exponents. Try this one. Leave your answers with positive exponents.
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Exponents
When fractions are to a negative power, a short cut is to flip the fraction and make the exponent positive.
Try...
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Exponents
Two more examples. Leave your answers with positive exponents.
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94 Simplify and write with positive exponents:
A
B
C
D Solution not shown
Exponents
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95 Simplify and write without a fraction:
A
B
C
D Solution not shown
Exponents
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96 Simplify. Answer may be in any form.
A
B
C
D Solution not shown
Exponents
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97 Simplify. Answer may be in any form.
A
B
C
D Solution not shown
Exponents
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Radicals
Return to Table of Contents
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Goals and Objectives
Radicals
Students will be able to put problems in simplest radical form, as well as be able to add, subtract, multiply and divide radical expressions.
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Radicals
Why do we need this?Engineers and Scientists perform complicated
calculations. If a problem requires multiple operations and answers are rounded and reused for the next step,
what happens to the answer in the end? It is not as accurate as possible. In science, we round only at the end. This way, we can have more accurate answers.
Being able to work with radicals, keeping them in simplest radical form, will allow us to maintain exact
numbers as we work with problems.
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Radicals
= 4.898979485566356....
Since 24 is not a perfect square, you must round the decimal to make the number reasonable. Your answer is then not an exact number. Simplest radical form allows us to simplify
radicals, but keeps them as an exact number.
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Radicals
Putting radicals in simplest radical form.
Perfect square numbers {4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169...}
Use these numbers to factor the radical. Then simplify.
Try...
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98 Find:
Radicals
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100 Put in simplist radical form:
A
BCD Solution not shown
Radicals
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101 Simplify:
A
B
C
D Solution not shown
Radicals
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102 Simplify:
A
B
C
D Solution not shown
Radicals
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104 Simplify:
A
B
C
D Solution not shown
Radicals
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Simplify: 3x + 4x and 3x + 4y
Think about: and
Adding and subtracting radicals - relate it to what you know...Radicals
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Adding and subtracting radicals is the same as adding and subtracting terms with variables. If the roots do not match, you
cannot put them together.
Radicals
Try...
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Radicals
Can you simplify these?
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Radicals
Try...
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106 Simplify:
A
B
C
D Solution not shown
Radicals
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107 Put in simplest radical form and collect like terms:
A
B
C
D Solution not shown
Radicals
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Radicals
Multiplying Radicals
Rule: Radical times radical, whole number times whole number.
Work out:
**All answers must be left in simplest radical form.
. .
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Radicals
*Remember to leave your answers in simplest radical form
. .
Try...
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And...
Radicals
. .
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112 Work out:
A
B
C
D Solution not shown
Radicals
.
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113 Multiply:
A
B
C
D Solution not shown
Radicals
.
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114 Multiply:
A
B
C
D Solution not shown
Radicals
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116 Simplify:
A
B
C
D Solution not shown
Radicals
.
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Radicals
Dividing Radicals
Rules are the same for dividing radicals as with multiplying radicals: Radical divided by radical, whole number divided by whole number.
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Radicals
Dividing radicals is a bit tougher because each problem is different. You know you are done with a question when there is no radical in the denominator and any fraction is reduced.
Removing a radical from the denominator is called "rationalizing the denominator."
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Radicals
Again, each division problem is different. Try simplifying everything you can before rationalizing the denominator.
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Radicals
Variables work the same way. Try...
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117 Divide:
A
B
C
D Solution not shown
Radicals
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118 Simplify:
A
B
C
D Solution not shown
Radicals
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119 Rationalize the denominator and simplify:
A
B
C
D Solution not shown
Radicals
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120 Rationalize the denominator:
A
B
C
D Solution not shown
Radicals
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121 Simplify:
A
B
C
D Solution not shown
Radicals
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