Polynomials 5.1 Monomials

Polynomials5.1 Monomials

Objectives:• Students will multiply and divide monomials• Students will solve expressions in scientific notation

Many times when we analyze data we work with numbers that are very large. To simplify these large numbers, these numbers may be written using scientific notation or using exponents.

A monomial is an expression that is a number, a variable, or the product of a number and one or more variables.

Monomials cannot contain variables in denominators, variables with exponents that are negative, or variables under radical signs.

Monomials Non- Monomials

5b, -w, 23, 𝑥2,1

3𝑥3𝑦4 1

𝑛4 , 3 𝑥, 𝑥 + 8, 𝑎−1



Constants are monomials that contain no variables.

Coefficients are numerical factors of variables.

The degree of a monomial is the sum of all the exponents of the variables. The degree of 12𝑔7ℎ4 𝑖𝑠 11. The degree of a constant is 0.

A power is an expression of the form 𝑥𝑛. The word power is also used to refer to the exponent itself.

Negative exponents express the multiplicative inverse of a number. 𝑥−2 =1

𝑥2

Is 𝑥−2 a monomial? No, why?



For any real number 𝑎 ≠ 0 and any integer n, 𝑎−𝑛 =1

𝑎𝑛 𝑎𝑛𝑑1

𝑎−𝑛 = 𝑎𝑛.

2−3 =1

23 𝑎𝑛𝑑1

𝑧−5 = 𝑧5

For any real number a and integers m and n, 𝑎𝑚 ∙ 𝑎𝑛 = 𝑎𝑚+𝑛.

𝑥2 ∙ 𝑥3 = 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 = 𝑥2+3 = 𝑥5

For any real number 𝑎 ≠ 0, and integers m and n, 𝑎𝑚

𝑎𝑛 = 𝑎𝑚−𝑛.

𝑔7

𝑔4 =𝑔 ∙ 𝑔 ∙ 𝑔 ∙ 𝑔 ∙ 𝑔 ∙ 𝑔 ∙ 𝑔

𝑔 ∙ 𝑔 ∙ 𝑔 ∙ 𝑔= 𝑔7−4 = 𝑔3;

ℎ3

ℎ8 =ℎ ∙ ℎ ∙ ℎ

ℎ ∙ ℎ ∙ ℎ ∙ ℎ ∙ ℎ ∙ ℎ ∙ ℎ ∙ ℎ=

1

ℎ5 = ℎ−5

Show that 𝑥0 = 1;

𝑥5

𝑥5 = 𝑥5−5 = 𝑥0 and𝑥5

𝑥5 =𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥

𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥= 1 therefore 𝑥0 = 1



Power of a Power: 𝒂𝒎 𝒏 = 𝒂𝒎𝒏

𝑎3 2 = 𝑎3 ∙ 𝑎3 = 𝑎3+3 = 𝑎6

Power of a Product: 𝒂𝒃 𝒎 = 𝒂𝒎𝒃𝒎

𝑎𝑏 3 = 𝑎𝑏 ∙ 𝑎𝑏 ∙ 𝑎𝑏 = 𝑎 ∙ 𝑎 ∙ 𝑎 ∙ 𝑏 ∙ 𝑏 ∙ 𝑏 = 𝑎3𝑏3

Power of a Quotient: 𝒂

𝒃

𝒏=

𝒂𝒏

𝒃𝒏

𝑎

𝑏

4=

𝑎4

𝑏4 ; 𝑎

𝑏

−3=

𝑏

𝑎

3=

𝑏3

𝑎3



Very large and very small numbers written in standard notation can be written as scientific notation; in the form 𝑎 × 10𝑛, where 1 ≤ 𝑎 < 10 and n is an integer.

6,380,000 = 6.38 × 106 ; 0.000047=4.7 × 10−5

4 × 105 2 × 107 = 4 ∙ 2 × 105 ∙ 107 = 8 × 1012

2.7 × 10−2 3 × 106 = 2.7 ∙ 3 × 10−2 ∙ 106 = 8.1 × 104

Dimensional Analysis uses units with the numbers. If units are given, the answer must have units.

After the sun, Alpha Centauri C is the closest star to Earth which is 4 × 1016 meters away. How long does it take light from Alpha Centauri C to reach Earth?

𝑑 = 𝑟𝑡; 𝑤ℎ𝑒𝑟𝑒 𝑑 𝑖𝑠 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒, 𝑟 𝑖𝑠 𝑟𝑎𝑡𝑒, 𝑎𝑛𝑑 𝑡 𝑖𝑠 𝑡𝑖𝑚𝑒

𝑡 =𝑑

𝑟=

4×1016 𝑚𝑒𝑡𝑒𝑟𝑠

3.00×108 𝑚𝑒𝑡𝑒𝑟𝑠/𝑠𝑒𝑐≈ 1.33 × 108 𝑠𝑒𝑐 or 4.2 years

Bookwork: page 226; problems 18-54 even

Polynomials5.2 Polynomials

Objectives:• Students will add and subtract polynomials• Students will multiply polynomials

Shenequa wants to attend an out-of-state university where the tuition is $8820. The tuition increases at a rate of 4% per year. Polynomials can be used to represent this increase in tuition.

If r represents the rate of increase, then the tuition for the second year will be 8820(1+r).

The third year tuition is 8820(1 + 𝑟)2, or 8820𝑟2 + 17,640𝑟 + 8820 when expanded.

A Polynomial is a monomial or a sum of monomials.

The monomials that make up a polynomial are called the terms of the polynomial.

Remember we can collect like terms in polynomials.

A polynomial with three terms is called a trinomial, e.g. 𝑥2 + 3𝑥 + 1; while 𝑥𝑦 + 𝑧3 is a binomial.

The degree of the polynomial is the degree of the monomial with the greatest degree.



To simplify a polynomial means to perform the operations indicated and combine like terms.

3𝑥2 − 2𝑥 + 3 − (𝑥2 + 4𝑥 − 2)

3𝑥2 − 2𝑥 + 3 − 𝑥2 − 4𝑥 + 2

2𝑥2 − 6𝑥 + 5

2𝑥(7𝑥2 − 3𝑥 + 5)

14𝑥3 − 6𝑥2 + 10𝑥

We use the distributive property when multiplying polynomials.

3𝑦 + 2 5𝑦 + 4 = 3𝑦 ∙ 5𝑦 + 3𝑦 ∙ 4 + 2 ∙ 5𝑦 + 2 ∙ 4

15𝑦2 + 22𝑦 + 8

This is called the FOIL method; Firsts, Outers, Inners, and Lasts.



The Vertical Method can also be used.

3𝑦 + 2

× 5𝑦 + 4

12𝑦 + 8

15𝑦2 + 10𝑦

15𝑦2 + 22𝑦 + 8

Bookwork: page 231; problems 16-32 even, and 38-50 even

34x 52

30 + 4x 50 + 2

60 + 8

1500 + 200 + 00 + 0

1768

Polynomials5.3 Dividing Polynomials

Objectives:• Students will divide polynomials using long division• Students will divide polynomials using synthetic division

Remember long division: 3248 divided by 24.

3000 + 200 + 40 + 820 + 4

100

− 2000 + 400

1000 – 200 + 40

+ 50

− 1000 + 200

−400 + 40 + 8

− 20

− − 400 − 80

120 + 8

+ 5

− 100 + 20

20 + (−12) Remainder of 8

24=

1

3

Answer: 135 1

3



In lesson 5.1 we learned how to divide monomials. We can also divide a polynomial by a monomial.

Simplify 4𝑥3𝑦2+8𝑥𝑦2−12𝑥2𝑦3

4𝑥𝑦=

4𝑥3𝑦2

4𝑥𝑦+

8𝑥𝑦2

4𝑥𝑦−

12𝑥2𝑦3

4𝑥𝑦

= 𝑥2𝑦 + 2𝑦 − 3𝑥𝑦2

The division algorithm can be used to divide a polynomial by a polynomial.

Use long division to find 𝑧2 + 2𝑧 − 24 ÷ 𝑧 − 4

𝑧 − 4 𝑧2 + 2𝑧 − 24

𝑧

− 𝑧2 − 4𝑧

6𝑧 − 24

+ 6

− 6𝑧 − 24The remainder is zero.



Simplify: 𝑡2 + 3𝑡 − 9 5 − 𝑡 −1

𝑡2 + 3𝑡 − 9−𝑡 + 5

−𝑡

− 𝑡2 − 5𝑡

8𝑡 − 9

− 8

− 8𝑡 − 40

31

This is a division problem, isn’t it?

Answer: −𝑡 − 8 +31

5−𝑡

5.3 Dividing PolynomialsPolynomials


Lets see what synthetic division looks like:5𝑥3 − 13𝑥2 + 10𝑥 − 8 ÷ 𝑥 − 2

Step 1:

Step 2:

Step 3:

Step 4:

Step 5:

Write the coefficients of the dividend in descending order of the degree.

Write the constant to the left and bring down the first coefficient. The divisor must be in the form of 𝑥 − 𝑟.

5 − 13 10 − 8

5

Multiply the first coefficient by the constant, then add to the second coefficient.

2

10

-3

Multiply the sum by the constant, then add to the next coefficient.

-6

4

Continue until the remainder is determined. The numbers on the bottom row are the coefficients of the quotient. Start with the power of x that is one degree less.

8

0

5𝑥2 − 3𝑥 + 4

5.3 Dividing PolynomialsPolynomials


Synthetic division works only when the divisor is in the form 𝑥 − 𝑟. If the coefficient on the variable is not 1, the divisor must be rewritten.

(8𝑥4 − 4𝑥2 + 𝑥 + 4) ÷ (2𝑥 + 1)

Divide the numerator and denominator by the divisor’s coefficient.

(4𝑥4 − 2𝑥2 +1

2𝑥 + 2) ÷ (𝑥 +

1

2)

−1

24 0 -2

1

22

-2 1 1

2−

1

2

4 0 -1 1 3

2

4𝑥3 − 2𝑥2 − 𝑥 + 1 +

32

𝑥 +12

3

2÷ 𝑥 +

1

2=

3

2÷

2𝑥 + 1

2=

3

2∙

2

2𝑥 + 1

4𝑥3 − 2𝑥2 − 𝑥 + 1 +3

2𝑥 + 1Bookwork: page 236; problems 16-50 even

Polynomials5.4 Factoring Polynomials

Objectives:• Students will factor polynomials• Students simplify polynomial quotients by factoring

We have seen where factoring an expression can simplify the expression…

8𝑎3𝑏2 + 4𝑎2𝑏3 − 2𝑎𝑏

2𝑎𝑏=

2𝑎𝑏(4𝑎2𝑏 + 2𝑎𝑏2 − 1)

2𝑎𝑏= 4𝑎2𝑏 + 2𝑎𝑏2 − 1

Polynomials can be factored the same way using Factoring Techniques.

Number of Terms Factoring Technique Examples

Any number Greatest Common Factor (GCF) 𝑎3𝑏2 + 2𝑎2𝑏 − 4𝑎𝑏2 = 𝑎𝑏(𝑎2𝑏 + 2𝑎 − 4𝑏)

Two Difference of two SquaresSum of two CubesDifference of two Cubes

𝑎2 − 𝑏2 = (𝑎 + 𝑏)(𝑎 − 𝑏)𝑎3 + 𝑏3 = (𝑎 + 𝑏)(𝑎2 − 𝑎𝑏 + 𝑏2)𝑎3 − 𝑏3 = (𝑎 − 𝑏)(𝑎2 + 𝑎𝑏 + 𝑏2)

Three Perfect Square Trinomials

General Trinomials

𝑎2 + 2𝑎𝑏 + 𝑏2 = 𝑎 + 𝑏 2

𝑎2 − 2𝑎𝑏 + 𝑏2 = 𝑎 − 𝑏 2

𝑎𝑐𝑥2 + 𝑎𝑑 + 𝑏𝑐 𝑥 + 𝑏𝑑 = (𝑎𝑥 + 𝑏)(𝑐𝑥 + 𝑑)

Four or more Grouping 𝑎𝑥 + 𝑏𝑥 + 𝑎𝑦 + 𝑏𝑦 = 𝑥 𝑎 + 𝑏 + 𝑦 𝑎 + 𝑏= (𝑎 + 𝑏)(𝑥 + 𝑦)

Polynomials5.4 Factoring Polynomials

Objectives:• Students will factor polynomials• Students simplify polynomial quotients by factoring

Factor each expression.

3𝑥𝑦2 − 48𝑥 = 3𝑥(𝑦2 − 16) = 3𝑥(𝑦 + 4)(𝑦 − 4)

𝑐3𝑑3 + 27 = 𝑐𝑑 3 + 33 = 𝑐𝑑 + 3 𝑐2𝑑2 − 3𝑐𝑑 + 9

𝑚6 − 𝑛6 This could be a difference of squares or a difference of cubes.

Difference of squares should be done first to make the next step easier.

= 𝑚3 + 𝑛3 𝑚3 − 𝑛3 = 𝑚 + 𝑛 𝑚2 − 𝑚𝑛 + 𝑛2 𝑚 − 𝑛 𝑚2 + 𝑚𝑛 + 𝑛2

5𝑥2 − 13𝑥 + 6 Use the reverse FOIL method. The coefficients must be two numbers whose product is 5 ∙ 6 and whose sum is −13. −3 𝑎𝑛𝑑 − 10

5𝑥 − 3 𝑥 − 2


Polynomials5.5 Roots or Real Numbers

Objectives:• Students will simplify radicals• Students will use a calculator to approximate radicals

Does everyone know why we call it squaring a number or cubing a number, while the other powers do not have a ‘name’?

When we square a number, we find the area of a square.

a

a = 𝑎2

When we cube a number, we find the volume of a cube.

b

b

b

= 𝑏3



What are we doing when we square a number? Multiply that number by itself.

What about when we cube a number? Multiply that number by itself three times.

If division is the opposite of multiplication, can we do the opposite of squaring or cubing?

Yes, we call this, finding the roots of a number, e.g. square root, cubed root, etc.

Can we perform any algebraic operation to find the roots of numbers?

No, either we know the roots or we do not know. A calculator helps.

Exponents help greatly. What is the fourth root of 𝑎𝑏 4? ab

For any real numbers a and b, if 𝑎2 = 𝑏, then a is a square root of b.

Since 52 = 25, 5 is a square root of 25.

For any real numbers a and b, and any positive integer n, if 𝑎𝑛 = 𝑏, then a is an nth root of b.

Since 25 = 32, 2 is a fifth root of 32.



A new symbol, called the radical symbol, is used to indicate the nth root of a number.

𝑛

Some numbers have more than one real nth root.

For example, 36 has two square roots, 6 and -6.

When there is more than one real root, the nonnegative root is known as the principle root.

Only when indicated by an index, are we interested in the negative root.

− 𝑜𝑟 ±

If n is odd and b is negative, 𝑛

−𝑏, there will be no positive root. The principle root is negative.



n 𝒏𝒃 𝒊𝒇 𝒃 > 𝟎

𝒏𝒃 𝒊𝒇 𝒃 < 𝟎 𝒃 = 𝟎

even one positive, one negative root no real roots One real root , 0

odd one positive root, no negative roots no positive roots, one negative root

0

± 25𝑥4 = ± 5𝑥2 2

= ±5𝑥2

− 𝑦2 + 2 8 = − 𝑦2 + 2 4 2

= − 𝑦2 + 2 4

532𝑥15𝑦20 =

52𝑥3𝑦4 5

= 2𝑥3𝑦4

−9 n is even and b is negative

Therefore, this does not have a real root.



Remember, when no index is given, we are finding the principle root. If n is even, then the principle root is a positive number.

If the nth root of an even power results in an odd power, you must take the absolute value of the result.

−5 2 = −5

= −5 = 5

−2 6 = −23

= 8

If the result is an even power or you find the nth root of an odd power, there is no need to take the absolute value. Why?

Bookwork: page 248; problems 16-56 even; look at 58-62

Polynomials5.6 Radical Expressions

Objectives:• Students will simplify radical expressions• Students will add, subtract, multiply, and divide radical expressions

How do we simplify the expression 3 ∙ 5 ? 3 ∙ 52

= 3 ∙ 3 ∙ 5 ∙ 5

= 3 ∙ 5

therefore; 3 ∙ 5 = 15

For any real numbers a and b, and any integer n > 1, then…

If n is even and a and b are both nonnegative, 𝑛

𝑎𝑏 = 𝑛 𝑎 ∙𝑛

𝑏

If n is odd, 𝑛

𝑎𝑏 = 𝑛 𝑎 ∙𝑛

𝑏

Though these two rules ‘look’ the same, the second rule allows for negative vales of a and b.

16𝑝8𝑞7 = 42 ∙ 𝑝4 2 ∙ 𝑞3 2 ∙ 𝑞

= 4𝑝4 𝑞3 𝑞

However, for this to be defined 16𝑝8𝑞7

must be nonnegative; meaning, 𝑞must be nonnegative.



What about division?Consider,

49

9. This is a perfect square, so

49

9=

7

3

2=

7

3

For any real numbers a and b ≠ 0, and any integer 𝑛 > 1, then 𝑛 𝑎

𝑏=

𝑛 𝑎𝑛

𝑏

How do we know a radical expression is in simplest form?

• The index n is as small as possible.• The radicand contains no factors (other than 1) that are nth powers of an integer

or polynomial.• The radicand contains no fractions.• No radicals appear in the denominator.

Wait! What do you mean, no radicals in the denominator?

How do you get rid of radicals in the denominator?



To eliminate radicals from the denominator, we rationalize the denominator.

Remember when we have a fraction in the denominator we multiply by a form of 1 to eliminate the fraction.

We do the same thing with radicals.

𝑥4

𝑦5=

𝑥4

𝑦5=

𝑥2 2

𝑦2 2 ∙ 𝑦

=𝑥2

𝑦2 ∙ 𝑦

=𝑥2 2

𝑦2 2 ∙ 𝑦

∙𝑦

𝑦

=𝑥2 ∙ 𝑦

𝑦2

5 5

4𝑎=

55

54𝑎

∙

58𝑎4

58𝑎4

WHY?

=

540𝑎4

532𝑎5

=

540𝑎4

2𝑎



We now know that 2 ∙ 2 = 2, does 2 + 2 = 2 ?

Given:

1

1

c

What is c = ? From geometry, 𝑎2 + 𝑏2 = 𝑐2

𝑐 = 2

If we double the triangle, can 2 + 2 = 2 ?

1

1

cNo, because the hypotenuse MUST be longer than any one side.

2

2



Adding radicals is like adding monomials. We must combine like terms.

Like radical expressions are alike if…

The indices are alike. 3 𝑎𝑛𝑑3

3 𝑎𝑟𝑒 𝑛𝑜𝑡 𝑎𝑙𝑖𝑘𝑒

The radicands are alike. 4

5 𝑎𝑛𝑑4

5𝑥 𝑎𝑟𝑒 𝑛𝑜𝑡 𝑎𝑙𝑖𝑘𝑒

24

3𝑎 𝑎𝑛𝑑 54

3𝑎 𝑎𝑟𝑒 𝑙𝑖𝑘𝑒 𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛𝑠

2 12 − 3 27 + 2 48 = 2 22 ∙ 3 − 3 32 ∙ 3 + 2 22 ∙ 22 ∙ 3 = 4 3 − 9 3 + 8 3

= 3 3



(5 3 − 6)(5 3 + 6) = 25 32 + 30 3 − 30 3 − 36 FOIL

= 75 − 36

= 39

Notice that 𝑎 𝑏 + 𝑐 𝑑 𝑎𝑛𝑑 𝑎 𝑏 − 𝑐 𝑑 are conjugates. The product of conjugates is always a rational number. Conjugates are used to rationalize denominators.

1 − 3

5 + 3∙

5 − 3

5 − 3=

5 − 3 − 5 3 + 3

25 − 3

=8 − 6 3

22

=4 − 3 3

11Bookwork: page254; problems 16-48 even

Polynomials5.7 Rational Exponents

Objectives:• Students will write radical expressions using exponents.• Students will simplify expressions in radical or exponent form.

In lesson 5.5, we determined that squaring a number and taking the square root of a number are inverse operations.

Does this mean we can express a radical as an exponent? Yes!

𝑏12

2

= 𝑏12 ∙ 𝑏

12

= 𝑏12+

12

= 𝑏

This means that 𝑏1

2 is a number whose square is 𝑏. Therefore, 𝑏1

2 = 𝑏

For any real number b and for any positive integer n, 𝑏1

𝑛 =𝑛

𝑏, except when 𝑏 < 0 and n is even.



Evaluate each expression.

16−14 =

1

1614

=1

416

=1

424

=1

2

Is there another way to evaluate this expression.

Remember, exponent rules allow us to do many different things. Allowing us to use our strengths.

= 24 −14

= 24 −14

= 2−1

=1

2



Evaluate each expression.

24335 = 2433

15

=5

2433

=5

35 3

=5

35 ∙ 35 ∙ 35

= 3535

= 33

= 27

= 3 ∙ 3 ∙ 3

= 27



The last example leads to the following rule.

For any nonzero real number b, and any integers m and n, with 𝑛 > 1, 𝑏𝑚

𝑛 =𝑛

𝑏𝑚 =𝑛

𝑏𝑚

, except when 𝑏 < 1 and n is even.

How do we know a rational expression is simplified?

• No negative exponents.• No fractional exponents in the denominator.• Not a complex fraction.• The index of any remaining radical is the least number possible.


Polynomials5.8 Radical Equations and Inequalities

Objectives:• Students will solve equations containing radicals.• Students will solve inequalities containing radicals.

A computer chip manufacturer has determined that the cost to manufacture their chips is

𝐶 = 10𝑛2

3 + 1500. This formula has a radical in it.

Equations that have variables in the radicand are called radical equations. To solve this type of equation, isolate the radicand and then raise each side of the equation to the power of the index of the radical to eliminate the radical.

𝑥 + 1 + 2 = 4 𝑥 + 1 = 2 𝑥 + 12

= 22 𝑥 + 1 = 4 𝑥 = 3

We should always check our solution. Sometimes we will obtain a solution that does not satisfy the equation. This solution is called an extraneous solution.

𝑥 − 15 = 3 − 𝑥 𝑥 − 152

= 3 − 𝑥 2 𝑥 − 15 = 9 − 6 𝑥 + 𝑥

−24 = −6 𝑥 4 = 𝑥 42 = 𝑥 2 16 = 𝑥



Lets check this solution. 𝑥 − 15 = 3 − 𝑥 16 − 15 = 3 − 16

1 ≠ −1

If we graph 𝑦 = 𝑥 − 15 and 𝑦 = 3 − 𝑥 on our calculators, we see the two graphs do not intersect; meaning, there is no solution.

3 5𝑛 − 113 − 2 = 0 3 5𝑛 − 1

13 = 2 5𝑛 − 1

13 =

2

3

5𝑛 − 113

3

=2

3

35𝑛 − 1 =

8

275𝑛 =

35

27

𝑛 =7

27If we check this solution, we find

7

27is a solution.



Knowing this information, we can solve radical inequalities. A radical inequality is an inequality with a radicand.

2 + 4𝑥 − 4 ≤ 6 First, the radicand must be greater than or equal to zero.

4𝑥 − 4 ≥ 0 𝑥 ≥ 1

Now we must solve the original inequality.

2 + 4𝑥 − 4 ≤ 6 4𝑥 − 4 ≤ 4 4𝑥 − 4 ≤ 16 𝑥 ≤ 5

It appears our solutions are 1 ≤ 𝑥 ≤ 5. By solving for f(0), f(2), and f(7) we can verify this.



To solve radical inequalities, use the following steps…

Step 1: if the index of the root is even, identify the values of the variable for which the radicand is nonnegative.

Step 2: solve the inequality algebraically.

Step 3: test values to check the solution or solution set.


Polynomials5.9 Complex Numbers

Objectives:• Students will add and subtract complex numbers.• Students will multiply and divide complex numbers.

When we solve the equation 2𝑥2 + 2 = 0, we find that 𝑥2 = −1. This is not a real solution; however, many solutions of radicands have a negative solution.

Rene Descartes proposed that the number i be defined such that 𝑖2 = −1.

This means that 𝑖 = −1. This is called the imaginary unit.

Numbers in the form of 3i, -5i, and i 2 are pure imaginary numbers.

Pure imaginary numbers are square roots of negative real numbers.

−𝑏2 = 𝑏2 ∙ −1 = 𝑏𝑖

−18 = −1 ∙ 32 ∙ 2

= 3𝑖 2

−125𝑥5 = −1 ∙ 52 ∙ 𝑥4 ∙ 5𝑥

= 5𝑖𝑥2 5𝑥



−2𝑖 ∙ 7𝑖 = −14𝑖2

= −14 ∙ −1

= 14

−10 ∙ −15 = 𝑖 10 ∙ 𝑖 15

= 𝑖2 150

= −1 ∙ 25 ∙ 6

= −5 6

𝑖45 = 𝑖 ∙ 𝑖44

= 𝑖 ∙ 𝑖2 22

= 𝑖 ∙ −1 22

= 𝑖

3𝑥2 + 48 = 0

3𝑥2 = −48

𝑥2 = −16

𝑥 = ± −16

𝑥 = ±4𝑖



What about the expression 5 + 2𝑖. Since 5 is a real number and 2i is an imaginary number, the terms are not like terms. This expression is called a complex number.

A complex number is any number that can be written in the form 𝒂 + 𝒃𝒊, where a and b are real numbers and i is the imaginary unit. a is the real part and b is called the imaginary part.

Two complex numbers are equal only if their real parts are equal and their imaginary parts are equal.

2𝑥 − 3 + 𝑦 − 4 𝑖 = 3 + 2𝑖 Can we solve this when we have two variables and one equation?

Yes, because x is a real part and y is an imaginary part; giving us two variables and two equations.

2𝑥 − 3 = 3 𝑎𝑛𝑑 𝑦 − 4 = 2

𝑥 = 3 𝑎𝑛𝑑 𝑦 = 6



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 the impedance 7 − 5𝑗 ohms.

𝐸 = 𝐼 ∙ 𝑍 𝐸 = (1 + 3𝑗) ∙ (7 − 5𝑗) 𝐸 = 7 − 5𝑗 + 21𝑗 − 15𝑗2

= 7 + 16𝑗 + 15 = 22 + 16𝑗 𝑣𝑜𝑙𝑡𝑠

Two complex number in the form of 𝑎 + 𝑏𝑖 𝑎𝑛𝑑 𝑎 − 𝑏𝑖 are complex conjugates. The product of complex conjugate is always a real number. This fact can be used to simplify the quotient of two complex numbers.

3𝑖

2 + 4𝑖=

3𝑖

2 + 4𝑖∙2 − 4𝑖

2 − 4𝑖=

6𝑖 − 12𝑖2

4 − 16𝑖2=

6𝑖 + 12

20=

3

5+

3

10𝑖

Remember, 𝑎 + 𝑏𝑖 is standard form for imaginary numbers.


Date post:	04-Dec-2021
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Polynomials 5.1 Monomials

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