Unit 4Unit 4Operations Operations
& Rules& Rules
Combine Like Terms
1) 3x – 6 + 2x – 8
2) 3x – 7 + 12x + 10
Exponent Rules
3) What is 2x 3x?
5x – 14
15x + 3
6x2
Warm up
Degree
The exponent for a variable
Degree of the Polynomial
Highest (largest) exponent of the polynomial
Standard Form
Terms are placed in descending order by the DEGREE
Write all answers in Standard Form!
Leading Coefficient
Once in standard form, it’s the 1st NUMBER in front of the variable (line leader)
# of Terms
Name by # of Terms
1 Monomial
2 Binomial
3 Trinomial
4+ Polynomial
Degree(largest
exponent)
Name by degree
0 Constant
1 Linear
2 Quadratic
3 Cubic
2 9y Special Names:
LinearBinomi
al
Degree Name:
# of Terms Name:
Leading Coefficient: -2
334xSpecial Names:
Cubic
Monomial
Degree Name:
# of Terms Name:
24 6x xSpecial Names:
Quadratic Binomi
al
Degree Name:
# of Terms Name:
Leading Coefficient: 4
3 27 2y y y Special Names:
CubicTrinomial
Degree Name:
# of Terms Name:
Leading Coefficient: 1
Adding Polynomial
s
2 22 4 3 5 1x x x x
1.
3x2 + x + 2
26 2 8x x 2.
x2 + 2x – 2
Subtracting
Polynomials
When SUBTRACTING polynomials
Distribute the NEGATIVE
2 23 10 8a a a a
3a2 + 10a – 8a2 + a
– 5a2 + 11a
3.
2 23 2 4 2 1x x x x
3x2 + 2x – 4 – 2x2 – x + 1
x2 + x – 3
4.
Multiplying
Polynomials
-2x(x2 – 4x + 2)
3 22 8 4x x x
5.
(x + 3) (x – 3)
2 9x
6.
(3x – 1)(2x – 4)
26 14 4 x x
7.
8. Find the area of the rectangle.
228 96 80 x x
7 10x
4 8x
9. Find the volume.
3 29 18 x x x
3x
6x
x
Warm upWarm upFind an expression for the area of
the following figure:
ChallengeChallengeFind an expression for the volume of
cylinder:
3 2( 8 12)x x x
Polynomial OpsQ8 of 20
Multiply:
2( 6)x
Find the perimeter
Find the Perimeter:
Polynomial OpsQ11 of
20
Find an expression for the volume of cylinder:
Polynomial OpsQ14 of
20
Write an expression for the volume of the box:
Polynomial OpsQ16 of
20
Find the area of the label.
Skills Skills CheckCheck
Square Roots and
Simplifying Radicals
Parts of a radicalParts of a radical
No number where the index is means it’s a square root (2)
index radicand
1. No perfect square factors other than 1 are under the radical.
2. No fractions are under the radical.
3. No radicals are in the denominator.
You try!
1.
20
52
You try!
2.
3 50
15 2
You try!
3.
120
2 30
Variables Under Square Roots
Even Exponent –
ODD Exponent –
Take HALF out (nothing left under the radical)
Leave ONE under the radical and take HALF of the rest out
13 24a b6 12a b a
18 5 4c d
3 22 2c d c
NNthth Roots Roots & Rational & Rational ExponentsExponents
Parts of a radical
No number where the root is means it’s a square root (2)
root radicand
Simplifying Radicals
Break down the radicand in to prime factors.Bring out groups by the number of the root.
root radicand
Simplify31. 128 3 2 2 2 2 2 2 2
34 2
Simplify
x3 32. 27x x x 3 3 3 3
x3
Simplify
x4 73. 324 2 2 2 2 2 x x x x x x x
x x 4 32 2
Simplify
x934.
27x
3
3
Rewriting a Rewriting a Radical to have Radical to have
a Rational a Rational ExponentExponent
Rewriting Radicals to Rational Exponents
Power is on topRoots are in the ground
powerpower
root rootradicand radicand
Rewriting Radicals to Rational Exponents
Power is on topRoots are in the ground
powerpower
root rootradicand radicand
Rewrite with a Rational Exponent
w5. 10 w1210
Rewrite with a Rational Exponent
p36. 7 p137
Rewrite with a Rational Exponent
5
7. 17 5217
Rewrite with a Rational Exponent
y288. y28
y14
Rewrite with a Rational Exponent
z3 69. z63
z 2
Rewriting Rational Exponents to Radicals
power power
rootrootradicand radicand
Rewrite with a Rational Exponent
(don’t evaluate)3510. 12
35 12
Rewrite with a Rational Exponent
(don’t evaluate)
2311. 13
23 13
Rewrite with a Rational Exponent
(don’t evaluate)
x3212. x
3
1 i 1. 36
2. 96
3. 325
4. 8575
6i4 6i5 13i35 7i
Warm Warm upup
Powers of iand Complex Operations
““I one, I one!!”I one, I one!!”Negatives in the middle.Negatives in the middle.
1
1
i
i
1
2
3
4
i
i
i
i
75
29
251
9536
5.
6.
7.
8.
i
i
i
i
Try these!
iii
1
Add and Add and Subtract Subtract Complex Complex NumbersNumbers
Add/Subt Complex Add/Subt Complex NumbersNumbers
1.Treat the i’s like variables2.Combine the real parts then
combine the imaginary parts
3.Simplify (no powers of i higher than 1 are allowed)
4.Write your answer in standard form a + bi
Simplify
9. (3 2 ) (7 6)i i
10 8i
Simplify
10. (6 5) (1 2 )i i 6 5 1 2i i 5 7i
Simplify
11. (9 4 ) (2 3 )i i
9 4 2 3i i 7 7i
Simplify
12. 9 (10 2 ) 5i i
9 10 2 5i i 1 7i
Simplify
4 3 4 313. (11 4 ) (2 6 )i i i i 4 3 4 311 4 2 6i i i i
111 4 2 1 6i i
9 10i
Multiplying Multiplying Complex Complex NumbersNumbers
Multiplying Complex Numbers
1.Treat the i’s like variables2.Simplify all Powers of i
higher than 13.Combine like terms
4.Write your answer in standard form a + bi
Multiplying Complex Numbers
14. (3 )i i 23i i
3 ( 1)i 1 3i
Multiplying Complex Numbers
15. (2 3 )( 6 2 )i i 212 4 18 6i i i
12 22 6( 1)i 12 22 6i 6 22i
Multiplying Complex Numbers
16. 3 8 5i i 224 15 8 5i i i
24 15 8 5 1i i 24 7 5i 29 7i
Dividing Dividing Complex Complex NumbersNumbers
What is a What is a Conjugate?Conjugate?
17. Dividing – Multiply top & bottom by the Conjugate
3 42 4
ii
2 4
2 4
i
i
2
2
6 12 8 164 8 8 16
i i ii i i
16 12 8 16
14 8 8 16
i i
i i
10 2020
i
10 2020 20
i
12
i