unit 1 polynomials notes - new paltz middle school · unit 1 polynomials notes 8 expressions vs...

UNIT 1 Polynomials NOTES

1

Adding & Subtracting Polynomials (Combining Like Terms)

Adding

Polynomials

Subtracting Polynomials

(if your 2nd polynomial is inside a set of parentheses).

(3x2 – 8x + 2) + (-5x2 -3x – 7)

(3x2 – 8x + 2) + (-5x2 - 3x – 7) ◊ ∂ ∞ ◊ ∂ ∞

3x2 – 8x + 2

+ -5x2 - 3x – 7

-2x2 - 11x – 5 Final answer

(8m2 – 5m + 2) - (-10m2 +7m – 6)

(8m2 – 5m + 2) - (-10m2 +7m – 6)

(8m2 – 5m + 2) + (+10m2 -7m + 6)

(8m2 – 5m + 2) + (+10m2 -7m + 6) ◊ ∂ ∞ ◊ ∂ ∞

8m2 – 5m + 2

+ 10m2 -7m + 6

18m2 –12m + 8 Final answer

FIRST, Identify the like terms.

Like terms

have matching bases

that have matching exponents.

THEN, Add the coefficient.

Yes, you still need to know your

integer rules.

BUT, Keep the matching bases

AND

Keep the matching exponents.

Before you do anything else,

DISTRIBUTE THE SUBTRACTION SIGN.

(THINK OF IT AS AN OPPOSITE SYMBOL)

Now, just follow the adding rules


2

LAWS OF EXPONENTS: work for Multiplication & Division

Positive Exponents Negative Exponents Zero Exponents

1 is what you start with. Then, it is repeatedly being MULTIPLIED by the base.

1 is what you start with. Then, it is repeatedly being DIVIDED by the base.

1 is what you start with. NOTHING IS DONE TO IT. So, 1 is what you end up with.

2+5

1 2+5

1 2 2 2 2 2 (1 multiplied by “2” “5” times)

32

2-5

52

1

22222

1

(1 multiplied by “2” “5” times)

32

1

20

1 (1 is NOT MULTIPLIED or DIVIDED by 2 at all !)

Any Base Raised to the 0 power is equivalent to 1.

EXCEPTION to the Rule 00 is “undetermined”.

Check it out in your calculator ,

Multiplying Powers

With the Same Base

Dividing Powers With the Same Base

Raising a Power To Another Power

Raising a Fraction To a Power

Raising a Product To a Power

Keep the common base. Add the exponents.

Keep the common base. Subtract the exponents.

Keep the common base. Multiply the exponents.

Raise the numerator to the power.

Raise the denominator to the power.

Raise EACH FACTOR to the power.

35 37

3 5 + 7

312

3

8

4

4

= 48-3 = 45

(53)6

= 5 36

= 5 18

2

22

5

3=)

5

3(

=25

9

(3 4)2

= 32 42

= 9 16


3

LAWS OF EXPONENTS: Special cases with signed coefficients

Positive Coefficient (with or without parentheses)

Negative Lead Coefficient without parentheses

Negative Lead Coefficient with parentheses

POSITIVE 1 is what you start with. Then, it is repeatedly being Multiplied by the base of +2.

NEGATIVE 1 is what you start with. Then, it is repeatedly being Multiplied by the base of +2.

POSITIVE 1 is what you start with. Then, it is repeatedly being Multiplied by the base of -2.

24

1 24

1 2 2 2 2

16

-24

-1 24

-1 2 2 2 2

-16

(-2)4

1 (-2)4

1 -2 -2 -2 -2

1 4 -2 -2

1 -8 -2

1 16

16

NOTE: If the negative sign is inside the parentheses and the exponent is an even number, the answer will be positive (as shown above). If the exponent is an odd number, then the answer will be negative.

}

}

}


4

When Final Answers Require MORE Simplification. Final Answers Should Have Positive Exponents

Answers that have Zero Exponents

When your answer results in a “zero exponent”, remember to finalize your answer.

It will equal 1.

Ex: 4

4

3

3 = 34--4 = 30 = 1

Answers that have Negative Exponents When your answer results in a “negative exponent”,

remember to finalize your answer. Convert to an equivalent fractional form (division problem)

with a “positive exponent.

5

2

5

5 = 52-5 = 5-3 =

35

1

NEGATIVE exponents in the denominator should be simplified further as well! Reverse the numerator and the denominator.

Then, make the exponent positive.

Ex: 3

1

8 = 83


5

How Can 30 be equivalent to 1???? This is an Alternate Way To Think About It, That Might Help You Believe It. Here is what it looks like expanded out:

Ex: 4

4

3

3

= 3•3•3•3

3•3•3•3

= 1•1•1•1

1•1•1•1

1•1•1•1 = 1

But…we know that 4

4

3

3

is also equal to 34--4 = 30

If 4

4

3

3

= 30 and 4

4

3

3

= 1

Then, 30 must equal 1.

How Can 5-3 be equivalent to 35

1

???? This is an Alternate Way To Think About It, That Might Help You Believe It. Here is what it looks like expanded out:

Ex: 5

2

5

5

= 5•5•5•5•5

5•5

= 5•5•5•5•5

5•5

5•5•5•1•1

1•1

= 35

1

But…we know that 5

2

5

5

is also equal to 52-5 = 5-3

If 5

2

5

5

= 5-3 and 5

2

5

5

= 35

1

Then, 5-3 must equal 35

1


6

Multiplying & Dividing Monomials Part Old-Fashioned Multiplying & Dividing

Part Laws of Exponents

Multiplying Monomials Dividing Monomials

(3a4b5c7)(-5a1b3c2)

3-5a4a1b5b3c7c2

-15a5b8c9 Final simplified answer

673

2105

cba5

cba20

6

2

7

10

3

5

c

c

b

b

a

a

5

20

4a2b3c-4

4

32

c

ba4

Final simplified answer

The Commutative Property

allows us to rearrange a

multiplication problem to a

form to works better for us.

Coefficients near Coefficients

Like Bases near Like Bases

Multiply the

coefficients just as

you would any

“normal” numbers.

Yes you will need to

know your integer

rules.

To multiply same

bases, you add the

exponents.

Think of this as several

little division problems.

Divide the

coefficients just as

you would any

“normal” numbers.

Yes you will need to

know your integer

rules.

To divide same bases,

you subtract the

exponents.

Remember, when you

have negative

exponents in your

final answer, you need

to rewrite using

positive exponents in

your final step.

c-4 really means

repeated division by

c.


7

To Multiply Polynomials just DISTRIBUTE: Example: Multiply 3x2 (4x3 + 5x - 2) 3x2 (4x3 + 5x - 2)

(3x2 4x3) + (3x2

5x ) + ( 3x2 -2)

Answer:

Dividing POLYNOMIALS by Monomials: 1) Divide each term of the polynomial by the monomial. 2) Hints about your answers: * If you are dividing a trinomial, your answer will be a trinomial. * If you are dividing a binomial, your answer will be a binomial. * If you are dividing a monomial, your answer will be a monomial.

Example: Divide (12x5 + 15x3 - 6x2) (3x2)

The other way to write a division problem.

12x5 + 15x3 - 6x2

Notice the long fraction bar means all terms 3x2 are being dividing by 3x

2.

Multiply 3x2 by

each term inside the

parentheses.

12x5 + 15x3 - 6x2

My Division Work

2

2

2

3

3x

6x -

3x

15x +

x3

x122

5

4x3 + 5x - 2 Answer


8

Expressions VS Polynomial Expressions

Expression: Numbers, symbols and operations (such as +, -, × and ÷) grouped

together that show the value of something.

A) Term of an expression: A number, a variable, or a product or quotient of numbers and variables. Notice “single” terms do not contain addition or subtraction.

12 q 11ab 4x3 3

-1 xyz5 w5

Terms are added/subtracted in algebra expressions. Each time you encounter an addition/subtraction symbol this is how you know that you are starting a new term.

Example 1: 11ab + 12 Example 2: 11ab - 12 ( 11ab is the 1 st term and 12 is the 2 nd term. ) ( 11ab is the 1 st term and - 12 is the 2 nd term. )

The examples above are both considered to be expressions with two terms.

Terms can be categorized as Like terms or Unlike terms.

1) Like terms: Terms that contain exactly the same variables, with each matching variable containing the same exact exponent.

Examples: 12 and 7 (All constants are considered like terms.) 5y and 2y 9ab and

52 ab

x2y3 and 4x2y3

2) Unlike terms: Terms that contain different variables or terms that have the same variables, but different exponents.

Examples: 5x and 2y 9ab and

52 a

x3y2 and 4x2y3

B) Coefficient of a term: A number before a variable. This constant tells how many of a variable you have. Ex: 5a really means a + a + a + a + a

NOTE: Like terms can have different coefficients, but the variable/exponent combinations must match exactly.

In the example 1x2y3 and 4x2y3 listed above, the x2y3 represents the “same quantity” in both terms. There is 1 set of x2y3 and another 4 sets of x2y3. Therefore, it makes logical sense to combine them and just say there are 5 sets of x2y3.

C) Fully Simplified Expressions: In order to be simplified an expression must only contain monomial terms, separated by + or – signs.

There cannot be any…. Like terms that still need to be combined by addition or subtractions.

Example: 4a + 5a is not simplified, until you combine like terms to 9a.

Laws of exponents that can still be applied (multiplying same bases, dividing same bases, power to power, changing an x0 to 1, and converting negative exponents.) Example: 4a2b3a5 is not fully simplified, because the a’s can be joined by adding the exponents. The expression simplifies to 4a7b3.

Factors that have not been “distributed”. Example: 8x(4x – 5) is not simplified fully. The 8x is a monomial factor and the 4x - 5 is a binomial factor. The 8x needs to be distributed (multiplied) into the 4x - 5, to simplify the expression to 32x2 – 40x.


9

Polynomial Expression: Numbers, symbols and operators (such as +, -, × and ÷) grouped

together that show the value of something. It is “almost” the same as a basic expression, but with one exception…

In simplified form, Polynomials expressions cannot contain division by a variable,

even though a “basic” expression can.

A) Categorizing Polynomials: A polynomial can be categorized by the number of unlike terms it contains. Monomial Binomial Trinomial (One Term) (Two Unlike Terms) (Three Unlike Terms) 3 7x + 5 4x + 2c + 7 2x 8x + 7y a + 2b + 4c 9z2 7a2 + bc x2 + 2xy + y2

Terms vs Factors in an Expression: Terms: Remember that individual terms are numbers, variables, or a product or quotient of numbers and variables, where each new term is separated by a (+ or -).

Factors: Numbers that are multiplied together. Ex) 5 • 4 (5 and 4 are the factors, since they are being multiplied.) Expressions that are multiplied together. Ex) 5a(a2 - 4)

(The factors are the monomial 5a and the binomial a2 – 4 since these expression are being multiplied.)

More examples of terms vs. factors

Polynomial Term OR

Polynomial Expression

Basic Term OR

Basic Expression (division by a variable)

12

q

11ab

4x3

3

1-xyz5

w5

2ba

5a-2

Remember: 5a-2 = 2

5

a,

which means you are dividing by a variable. So, it is not a polynomial.

3x2y4 + 8x5y2 Can be classified as a binomial

expression. Contains 2 terms, 3x2y4 and 8x5y2. The term 3x2y4 has factors of 3, x2,

and y4. The term 8x5y2, has factors of 8, x5,

and y2.

(2x + 3)(5x2 + 8x - 4) This is a polynomial, but it is not simplified,

because it is in factored form. The factors are a binomial 2x + 3 and a

trinomial 5x2 + 8x – 4. The factor 2x + 3 has 2 terms 2x and 3. The term 2x in the factor 2x - 3 has it’s

own factors of 2 and x. Etc, etc…


10

Polynomials: Standard Form and Degree

A) Polynomials with ONE Variable (Degree and Standard Form)

1) The degree corresponds to term with highest exponent on a variable.

Example 1: 5x2 + 7x – 3 (which equals 5x2 + 7x1 – 3x0)

The degree of this polynomial is 2.

2) Standard Form: The exponents in the terms of the polynomial should be in descending

order.

Example 2: -4x3 + 3x5 + 2x becomes

3x5 - 4x3 + 2x

B) Polynomials with MORE than one Variable (Degree and Standard Form) 1) The degree corresponds to the term in which the sum of the exponents on

the variables is the largest.

Example 1 5y3z5 + 4x1z9 - 8x2y6 – 3xy Sum of exponents Sum of exponents Sum of exponents Sum of exponents

8 10 8 2

The degree of this polynomial is 10.

2) Standard Form: The sum of the exponents in the terms of polynomial should be in

descending order. If there are terms with the same degree, then the term that has a

variable that comes first in the alphabet should come first.

Example 2: 5y3z5 + 4x1z9 - 8x2y6 – 3xy Sum of exponents Sum of exponents Sum of exponents Sum of exponents

8 10 8 2

becomes

4x1z9 - 8x2y6 + 5y3z5 – 3xy

Sum of exponents Sum of exponents Sum of exponents Sum of exponents 10 8 8 2

Reminder: A polynomial should be fully simplified before placing it in standard form. (see your prior notes for a reminder).

Both terms have a degree of 8, but, x comes before y in the alphabet,

so 8x2y6 is written 1st.


11

DOUBLE DISTRIBUTE:

Example: Multiply (3x – 5)(2x + 4) 1

st, set up 2 single distribution expressions.

2nd

, distribute the 3x into 2x - 4. 3

rd, distribute the -5 into 2x - 4.

4th, combine any like terms created.

3x(2x + 4) + -5(2x + 4)

6x2 + 12x + -10x – 20

FINAL Answer 6x2 + 2x – 20

Alternate Technique

3x – 5) (2x + 4)

FIRSTS 6x

2

OUTER 12x

INNER

-10x

LASTS

-20

F.O.I.L. Technique

FIRST (F): Multiply the FIRST TERMS in each pair of binomials.

OUTER (O): Multiply the OUTER TERMS in each pair of binomials.

INNER (I): Multiply the INNER TERMS in each pair of binomials.

LASTS(L): Multiply the LAST TERMS in each pair of binomials.

Combine Like Terms

to get the final answer:

6x2 + 12x -10x - 20 6x2 + 2x – 20


12

Greatest Common Factor (GCF) :

The greatest common factor of 2 or more numbers is the largest factor the numbers have in common.

The GCF of 12 and 18 is 6.

62 = 12

63 = 18

Although they have other common factors, 6 is the largest.

The greatest common factor of 2 or more powers is the largest power each base has in common.

The GCF of x5 and x3 is x3.

x x x x x = x5

x x x = x3

Notice when you expand each power they have 3 factors of x in common,

which is x3 and this is your GCF.

You will be asked to find the GFC for terms that have a combination of coefficients (numbers) and powers.

Find the GCF of each part separately, then put each part together to create the complete GCF.

Example: Find the GCF of 12x5 and 18x3 The GCF of 12 & 18 is 6 The GCF of x

5 & x

3 is x3

So, the GCF of 12x5 and 18x

3 is 6x3

Find the GCF of 30m6 and 40m8 The GCF of 30 and 40 is 10

The GCF of m6 and m

8 is m6

So, the GCF of 30m6 and 40m

8 is 10 m6

NOTICE: In each case the GCF was the base with the smallest power.

For example: For x5 & x3 the GCF was x3.

For m6 & m8 the GCF was m6. .


13

Observations When Double Distributing

That Help You Understand Reverse Distribution (Factoring)

(3x – 1)(2x + 3)

3x(2x + 3) + -1(2x + 3)

6x2 + 9x + -2x -3

6x2 + 9x + -2x -3

6x2 + 7x -3

-18x2

Product of the Outer Terms

-18x2

Product of the Inner Terms

Notice: If you check mid-way through the process, you will notice that…

The product of the inners

EQUALS

The product of the outers

7x

Sum of the Inner Terms

Inner Term in the Final Trinomial is also

7x

Notice: If you check at the end of the process, you will notice that…

The sum of the inners

EQUALS

The sum of the final middle term of the trinomial

CONCLUSION: To “reverse distribution” (factor a trinomial)

You must find a pair of unique numbers. The 2 numbers must… Have a product that is equal the product of the outer terms of the trinomial.

But the same 2 numbers must also, Have a sum that is equal to the middle term of the trinomial.

WE WILL PRACTICE THIS TECHNIQUE IN THE NEXT LESSON


14

Product

-30

-2 15 Sum

13

Split 13x into

-2x and +15x,

using the -2 & 15,

since it has the

correct product/sum.

You will now have 4

terms instead of 3.

Reversing DOUBLE DISTRIBUTION:

FACTORING a Quadratic Expression by…. Finding the PRODUCT / SUM

To SPLIT THE MIDDLE

ax2+ bx + c

10x2 + 13x – 3

10x2 - 2x + 15x – 3

2x(5x - 1) + 3(5x - 1)

(2x + 3)(5x - 1)

4th: Factor the GCF from first half of the expression.

Factor the GCF from last half of the expression.

5th: Rewrite the expression as 2 Binomials. The 2 GCF’s that you factored out form the first binominal.

The common binomial that remains forms the 2nd

binomial.

1st: Determine the product of the a & c coefficients. (-30).

Identify the b coefficient (13).

Product of -30

Sum of 13

2nd: Determine 2 numbers that have a

product of “-30”.

But, the same pair of numbers must also have a sum of “13”

ONLY 1 UNIQUE PAIR WILL WORK. In this case, that pair is -2 & 15.

Factor by GCF Factor by GCF

3rd: Split the b-term 13x into 2 terms using the pair of numbers

you just found. The term 13x will become -2x and +15x .

FINAL ANSWER

Create a binomial,

using the GCF’s

of 2x & 3.

Create a 2nd

binomial, by using

the common

binomial 5x -1.


15

CHECKING YOUR ANSWER

“BY HAND”

Since Factoring is the “reverse” of distribution, Distribution is the reverse of factoring!

So, check your Factored answer with Distribution.

(2x + 3)(5x - 1)

Double Distribution-To Check

2x(5x – 1) + 3 (5x – 1) 10x

2 – 2x + 15x – 3

10x2 + 13x – 3

10x2 + 13x – 3

Since this is the original Quadratric Expression

(before factoring), then our factoring is correct.

CHECKING YOUR ANSWER

“USING THE TI-84” The objective is to ensure the original expression is EQUIVALENT

to the new expression.

If the expressions are equivalent, then placing both in the TI-84, as

equations, SHOULD result in identical input(x)/output(y) charts. It

also, SHOULD result in identical graphed images.

TO CHECK:

1) Press . Place both expressions in your calculator.

2) Your screen should look like this when you are done.

HINTS:

For a variable x press…

Before you key in your exponent

press…

To move out of exponent mode

press…

3) Press to make sure the GRAPHS match.

Press to make sure Y1 and Y2 match.

The images

are the

same, so it

looks like

only 1 is

graphed .

The Y1 and

Y2 columns

are

identical,

which

implies the

expressions

are

equivalent.

.


16

Product

-10

2 -5 Sum

-3

FACTORING When the Lead “a” Coefficient is 1.

x2 -3x -10 Remember there is always an invisible Lead 1 coefficient

if no number is written. 1x

2 -3x - 10

SHORTCUT RULE

When the lead “a” coefficient is “1” the product/sum values -5 & 2 are the same values in your final answer.

This allows you to SKIP all the Split the Middle steps .

Go straight to your final answer and create your factors using the product/sum numbers.

BUT, BE CAREFUL!!! This shortcut only works when the lead coefficient is 1.

Product of -10

Sum of -3

(x + 2)( x - 5) Proceed Straight to the Final Answer

Use the Product/Sum Values to Create the Factors

NOTICE: Split the Middle is UNNECESSARY if your

Lead Coefficient is 1.

SO, SKIP all the Split the Middle Step!!!

1x2 -3x – 10 1x2 - 5x + 2x – 10

x(x - 5) + 2(x - 5)

(x + 2)( x - 5)


17

Special Cases Factoring by Split the Middle

Factoring by Split the Middle for a Degree 4 Trinomials

Degree 4 Trinomial

Ex: x4+ 9x2 + 18

x4+ 3x2 + 6x2 + 18 x2(x2 + 3) +6(x2 + 3)

(x2 + 6) (x2 + 3)

Solve just as you would with a Degree 2 Polynomial, except…

You will be splitting up x2 ’s instead of x’s. Your binomial factors with have x2 ’s instead of x’s.

Degree 2 Trinomial

Ex: x2+ 9x + 18

x2+ 3x + 6x + 18 x (x + 3) +6(x + 3)

(x + 6) (x + 3)


18

Difference of Perfect Squares (DOPS) In this case there is a binomial with the following characteristics.

BOTH terms are perfect squares.

The terms are separated by a subtraction sign.

You have 2 Options. Learn both. Each method will come in handy!

OPTION 1: Create a trinomial by placing a 0x into the binomial to create the middle term of the trinomial.

Then, solve as you usually would using product/sum AND split the middle.

Ex: x2 - 25

x2 +0x - 25

x2 + 5x -5x - 25 x(x + 5) -5(x + 5)

(x + 5) (x - 5)

OPTION 2: Square Root the first term x2. Use the square x

as the first term in each factored binomial.

Square Root the send term 25. Use the square root of 5 as the 2nd term of each factored binomial.

Place a + in the first binomial.

Place a – in the 2nd binomial.

Ex: x2 - 25

Square root of x2 is x. Square root of 25 is 5.

THE ANSWER is…

(x + 5) (x - 5)


19

unit 1 polynomials notes - new paltz middle school · unit 1 polynomials notes 8 expressions vs...

Documents