Download Module 3 notes -Polynomial A polynomial is an algebraic

Module 3 notes
-Polynomial
A polynomial is an algebraic expression composed of both variables
and constants that can be added, subtracted, multiplied, divided, or be
raised to whole number exponents. An addition or subtraction sign
separates each term in a polynomial. Each term in a polynomial is a
monomial.
-Monomial
Mono means one.
A monomial is an algebraic expression/polynomial consisting of one
single term.
Some examples of monomials are x, 5x3, -7sy, and 5.
-Binomials
Bi means two.
A binomial is an algebraic expression consisting of two terms. You
could also say that a binomial is an expression that is a sum or
difference of two monomials.
Some examples of binomials are (5x2 – 2), (x + 2), (2x + 1).
Trinomials
Tri means three.
A trinomial is an algebraic expression consisting of three terms. You
could also say that a trinomial is an expression that is a sum or
difference of three monomials.
Some examples of trinomials are (5x2 – 2x + 3), (x2 + 2x + 2), (2x2 –
x + 1).
To multiply binomials
Remember three key concepts.
-Key Concept #1
Multiplying involves the following:
A positive times a positive = a positive.
(+12) x (+2) = (+24)
A negative times a negative = a positive.
(-12) x (-2) = (+24)
A positive times a negative = a negative.
(+12) x (-2) = (-24)
-Key Concept #2
When multiplying two powers with the same base, add the exponents
and keep the base the same.
The rule states that na x nb = na+b.
Example: n3 x n2 = n2+3 = n5
Key Concept #3
When multiplying monomials, combine all the coefficients, and then
the variables. Use the rules above.
Example: Simplify 5x2 - 9x2.
Notice, these are like terms because the power in both terms is "x2".
The coefficient of each term includes its sign; in 5x2, the coefficient
is +5, while in -9x2, it is -9. This is very important to remember.
Combine the coefficients to simplify the expression. Multiplying Binomials
-When multiplying two binomials, we use the acronym FOIL to
remember the order. (An acronym is a word formed from the initials of
the words in a phrase; for example, NHL-stands for National Hockey
League.)
FOIL stands for First, Outside, Inside, Last.
First means that the first term in both binomials must be multiplied
together.
Outside means that the two outermost terms must be multiplied
together.
Inside means that the two innermost terms must be multiplied
together.
Last means that the last term of both binomials must be multiplied
together.
-After you have performed these multiplications, combine any like
terms. Like terms are two or more terms that have the same variables
raised to the same exponent. This involves adding or subtracting the
numerical coefficients and combining that with the variable.
Example: Multiply (x + 2)(x - 8).
Step 1: Multiply the first term in each binomial.
This step produces x2.
Step 2: Outside: Multiply the outermost terms.
This step produces -8x.
Step 3: Inside: Multiply the innermost terms.
This step produces +2x.
Step 4: Last: Multiply the last terms of each binomial.
This step produces –16.
Therefore: (x + 2) (x – 8) = x2 – 8x +2x – 16
= x2 – 6x – 16
Notice that –8x and +2x are like terms; therefore, after combining
those two terms, the resulting trinomial was x2 – 6x – 16.
Checking Your Answers
To check to see if you have simplified algebraic expressions correctly,
substitute numbers for the variables in both the original question and
the answer. If they have the same value, you have done the question
correctly.
-For example, check the following question.
Let the value of x be 2 (x = 2). In fact, using a number that is 2 or
greater is always a good technique. Sometimes 0 and 1 make your
solution seem correct even when it is not, because taking 0 or 1 to a
positive exponent does not change the value.
Example:
Because they both have the same values, your multiplication is correct.
-Multiplying polynomials
Multiplying polynomials is similar to multiplying binomials. One must
multiply every term in one polynomial with every term in the other
polynomial.
This multiplication
illustrates
the values,
distributive
property. is correct.
Because
they both have
the same
your multiplication
Example:
(x – 3)(x2 + 6x – 5)
Again, every term in one polynomial must be multiplied to every term
in the other.
Multiplying each step produces the following. (Notice that like terms
are stacked so that combining terms becomes easier.)
After the terms are combined from
Steps 2, 3, 4, and 5, the resulting
polynomial is x3 + 3x2 – 23x + 15.
Review
Factoring is rewriting an expression as a product of two or more
expressions: 2x(x + 1) is the factored form of 2x2 + 2x.
Greatest Common Factor (GCF) is the largest number or
expression that divides into a given expression without a
remainder.
The greatest common factor of 12 and 8 is 4 because 4 is the largest
number that will divide into both 12 and 8 without remainder.
The greatest common factor of 2x and 6xy is 2x because 2x is the
greatest term that will divide into 2x and 6xy without remainder.
Prime factorization represents a number as a product of its prime
factors: 96 is 2 x 2 x 2 x 2 x 2 x 3.
Powers are repeated multiplication: n3 means n x n x n.
Finding the Common Factor of a Polynomial
To find the common factor of a poynomial, follow these steps.
Step 1:
Factor fully each term in the polynomial. Ignore the sign of each term
for now. Include them when the polynomial is rewritten in factored
form. Factor coefficients into prime factors, and express any powers as
repeated multiplication.
Example: 14 x2y3 – 56xy2z2 + 28 xy2
14 x2y3 = 2 ● 7 ● x ● x ● y ● y ● y
56 xy2z2 = 2 ● 2 ● 2 ● 7 ● x ● y ● y ● z ● z
28 xy2 = 2 ● 2 ● 7 ● x ● y ● y
Step 2:
Examine each fully factored term, and circle the items that are
common to each term. In the example, the following circles show
factors that are common to each term.
Step 3:
Multiply the circled items together to get the GCF.
Step 4:
Rewrite the original polynomial 14 x2y3 – 56 xy2z2 + 28 xy2 in factored
form. Notice that the arithmetic symbols have been highlighted in red
to show that these do not change when the polynomial is in factored
form. To factor, follow this process:
This is what you get when you use the division rule. When you divide
exponents with the same base, you subtract the exponents. Also,
remember that anything taken to the zero exponent has a value of 1.
The above expression simplifies to the following:
Of course, anything multiplied by 1 is itself, and anything to the
exponent 1 can be written without actually showing exponent 1. In
other words, x1 = x. Therefore, the step above simplifies to
This is in factored form because the last operation is multiplication
(See below.) The two factors are (14xy2) and (xy – 4z2 + 2).
Step 5:
Check by multiplying the factors. If the product is equivalent to the
original polynomial, then your work is correct.
Notice, in the example above, a lot of detail was provided within each
step to eliminate confusion. As you practice, you will become more
proficient at putting polynomials in factored form, and you will need
fewer steps.
Factoring Trinimials
The next type of trinomial to be factored has the value of "a" not equal
to one, even after the GCF has been removed.
To factor ax2 + bx + c, where a ≠ 1, a factoring method called
grouping - decomposition is used. The steps used in this type of
factoring are the following:
Step 1: If possible, remove a GCF.
Step 2: Determine all the pairs of factors of the product "ac". Add
these factors together. The pair of factors that have a sum of "b" are
used in step 3.
Step 3: Rewrite the x term as the sum of the two terms with these
factors as coefficients. This step "decomposes" the middle term into
two pieces.
Step 4: Remove a common factor from the first two terms, and then
remove a common factor from the last two terms. This is the grouping
part of the method.
Step 5: There will be another common factor in the terms. Remove
the common factor and simplify. You have now factored the trinomial.
Example: Factor the following: 2x2 + 11x + 12.
Step 1: Remove any common factors.
Inspect 2x2, 11x and 12. There is no common factor.
Step 2: Determine the correct product (ac) and sum (b). Since a = + 2, c = +12, b =
+11, the value of ac is 24.
Find the two factors that have a product of ac = + 24 and a sum of b = +11.
Factors of +24
1 and 24
–1 and –24
2 and 12
–2 and –12
3 and 8
–3 and –8
4 and 6
–4 and –6
Product (a●c)
24
24
24
24
24
24
24
24
Sum (b)
25
–25
14
–14
11
–11
10
–10
Satisfies both conditions?
no
no
no
no
yes
no
no
no
From the chart, the two factors are 3 and 8 because they satisfy both conditions.
Step 3: Decompose the middle term.
2x2 + 11x + 12 becomes 2x2 + 8x + 3x + 12
Step 4: Group by taking a common factor out of first two terms (in green) and last
two terms (in pink) and then group.
2x2+8x + 3x + 12
GCF for 2x2 + 8x = 2x
GCF for 3x + 12 = 3
2x(x + 4) + 3(x + 4)
Notice that each term has (x + 4)(2x + 3). This is the factored form of 2x2 + 11x
+ 12.
Always test the answer by multiplying using FOIL. The result should be equal to
the original.
Because of this was the original polynomial, you know this trinomial has been
factored correctly.
-Special factors
A perfect square trinomial is the result of squaring a binomial. For
example, (x + 3)2 produces the trinomial x2 + 6x + 9.
Many students make the error of thinking that (x+3)2 = x2 + 9
because they do not realize (or they forget) that (x + 3)2 = (x + 3)(x
+ 3).
When this is multiplied, a middle term occurs.
The method of FOIL demonstrates this.
To factor a perfect square trinomial where a=1, follow the process
shown in the previous sections.
Example: Factor x2 –10x + 25
Because this is a simple trinomial where a=1, find two numbers that
multiply to give +25 and add to give –10.
_____ x _____ = +25
_____ +_____ = –10
The numbers are –5 and –5.
–5 x –5 = +25
–5 + –5 = –10
Rewrite this as factors to get (x – 5)(x – 5), which is (x – 5)2.
Sometimes, you need to factor a perfect square trinomial where a
does not equal 1. In these cases, recognize that it is a perfect square
trinomial meets the following criteria:
The first and third terms are perfect squares (example 4, 9,
16, 25, 36, 49, etc.)
The second term (b) is equal to the square root of the first
term a multiplied by the square root of the third term (C)
and then all multiplied by 2.
As an example: 16x2 + 24x + 9 is a perfect square because
16 and 9 are both perfect squares.
The square root of 16 (which is 4) times the square root of 9
(which is 3) and then all multiplied by 2 is 24. In other
words, Therefore, to factor a perfect square trinomial, you have two options:
Method A: Continue to factor as any trinomial where A is not equal to
1.
Method B: Recognize it to be a perfect square trinomial, and take a
short cut.
Both methods are shown below. You can choose which you prefer.
Example: Factor 16x2 + 24x + 9.
Method A: Factoring as a Trinomial
Start by trying to remove the GCF, in this case, there is no GCF.
Then, find two numbers that multiply to give a●c and add to give
b. (a = 16, b = 24, c= 9) a = 16 x 9 = 144
Factors of 144 are
(1 x 144), (2 x 72), (3 x 48), (4 x 36), (6 x 19), (8 x 18) and
(12 x 12).
The only factors that add to 24 are (12 x 12).
Write the trinomial as a polynomial with four terms in which the
middle term is decomposed: 16x2 + 12x +12x + 9 (Group like
terms.)
4x(4x + 3) +3(4x + 3) (Take out common factor.)
(4x
+ 3)(4x + 3) (Write as a perfect square.)
Therefore, 16x2 +
24x + 9 = (4x +3)2
Method B: Recognizing and Factoring as a Perfect Square
The challenge of this method is to recognize the polynomial as a
perfect square. Consider the factored forms of some perfect
squares that have been factored.
Check to see if there is a pattern. x2 + 16x + 64 = (x + 8)2
49x2 +
14x +1 = (7x + 1)2
x2 + 2x +1 = (x + 1)2
4x2 – 36x + 81 =
(2x - 9)2
x2 – 10 x + 25 = (x - 5)2
Have you figured out the pattern?
To change a perfect square trinomial into factored form... Factoring a Difference of Two Perfect Squares
-Also known as the difference of squares, the difference of perfect
squares occurs when a number is squared (multiplied by itself) and
then subtracted from another squared number.
Example: Factor 16x2 - 25
This expression is a difference of squares because 16x2 is a perfect
square, 25 is a perfect square, and the minus sign means difference.
When factoring a difference of squares, take the square root of each
term in the polynomial to find the terms of the binomials. One factor
always has a plus sign; the other has a minus sign.
To factor 16x2 – 25, follow the following steps:
Therefore the factored form of 16x2 – 25 is (4x + 5)(4x – 5).