# Solving Equations containing Binomial Expressions

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This pdf includes the following topics:-
Polynomials and Factoring
monomial
binomial
trinomial
polynomial
degree of a monomial
and many more.
1. Polynomials
Polynomials and
and
Factoring
Factoring
The basic building blocks of
algebraic expressions
2. The height in feet of
a fireworks launched
straight up into the air
from (s) feet off the
ground at velocity (v) after
(t) seconds is given by the
equation:
-16t2 + vt + s
Find the height of a
firework launched from a 10
ft platform at 200 ft/s
after 5 seconds.
-16t2 + vt + s
-16(5)2 + 200(5) + 10
=400 + 1600 + 10
610 feet
3. In regular math books, this is called
“substituting” or “evaluating”… We are given
the algebraic expression below and asked to
evaluate it.
x2 – 4x + 1
We need to find what this equals when we put a
number in for x.. Like
x=3
Everywhere you see an x… stick in a 3!
x2 – 4x + 1
= (3)2 – 4(3) + 1
= 9 – 12 + 1
= -2
4. What about x = -5?
Be careful with the negative! Use ( )!
x2 – 4x + 1
= (-5)2 – 4(-5) + 1
= 46
You try a couple
Use the same expression but let
x = 2 and
x = -1
5. That critter in the last slide is a polynomial.
x2 – 4x + 1
Here are some others
x2 + 7x – 3
4a3 + 7a2 + a
nm2 – m
3x – 2
5
6. For now (and, probably, forever) you
can just think of a polynomial as a
bunch to terms being added or
subtracted. The terms are just
products of numbers and letters
with exponents. As you’ll see later
on, polynomials have cool graphs.
7. Some math words to know!
monomial – is an expression that is a number, a
variable, or a product of a number and one or
more variables. Consequently, a monomial has no
variable in its denominator. It has one term.
(mono implies one).
13, 3x, -57, x2, 4y2, -2xy, or 520x2y2
(notice: no negative exponents, no fractional
exponents)
binomial – is the sum of two monomials. It has two
unlike terms (bi implies two).
3x + 1, x2 – 4x, 2x + y, or y – y2
8. trinomial – is the sum of three monomials. It has
three unlike terms. (tri implies three).
x2 + 2x + 1, 3x2 – 4x + 10, 2x + 3y + 2
polynomial – is a monomial or the sum (+) or The ending of these
words “nomial” is Greek
difference (-) of one or more terms. for “part”.
(poly implies many).
x2 + 2x, 3x3 + x2 + 5x + 6, 4x + 6y + 8
• Polynomials are in simplest form when they contain no like
terms. x2 + 2x + 1 + 3x2 – 4x when simplified
becomes 4x2 – 2x + 1
• Polynomials are generally written in descending order.
Descending: 4x2 – 2x + 1 (exponents of variables decrease
from left to right)
Constants like 12 are monomials
since they can be written as 12x0 =
12 · 1 = 12 where the variable is x 0.
9. The degree of a monomial - is the sum of the
exponents of its variables. For a nonzero
constant, the degree is 0. Zero has no degree.
Find the degree of each monomial
a) ¾x degree: 1 ¾x = ¾x1. The exponent is 1.
b) 7x2y3 degree: 5 The exponents are 2 and 3. Their sum is 5.
c) -4 degree: 0 The degree of a nonzero constant is 0.
10. Here’s a polynomial
2x3 – 5x2 + x + 9
Each one of the little product things is a “term”.
2x3 – 5x2 + x + 9
term term term term
So, this guy has 4 terms.
2x3 – 5x2 + x + 9
The coefficients are the numbers in front of the letters.
2x3 - 5x2 + x + 9
NEXT
2 5 1 9
We just pretend
this last guy has a
Remember
letter behind him.
x=1·x
11. Since “poly” means many, when there is only one term,
it’s a monomial:
5x
When there are two terms, it’s a binomial:
2x + 3
When there are three terms, it a trinomial:
x2 – x – 6
won’t go there, too hard to pronounce.
This guy is just called a polynomial:
7x3 + 5x2 – 2x + 4 NEXT
12. So, there’s one word to remember to classify:
degree
Here’s how you find the degree of a polynomial:
Look at each term,
whoever has the most letters wins!
3x2 – 8x4 + x5
This guy has 5
letters…
The degree is 5.
This is a 7th degree polynomial:
6mn2 + m3n4 + 8
This guy has 7 letters…
The degree is 7 NEXT
13. This is a 1st degree polynomial
3x – 2
This guy has 1
letter…
The degree is 1.
By the way, the
coefficients don’t
have anything to
This guy has no
do with the
degree.
8 letters…
The degree is 0.
How many letters does he have? ZERO!
So, he’s a zero degree polynomial
Before we go, I want you to know that Algebra
isn’t going to be just a bunch of weird words
that you don’t understand. I just needed to
14. 3x4 + 5x2 – 7x + 1 term
term term term
The polynomial above is in standard form.
Standard form of a polynomial - means that
the degrees of its monomial terms decrease
from left to right.
Name using
Polynomial Degree Name using Number of number of
Degree Terms terms
7x + 4 1 Linear 2 Binomial
2
3x + 2x + 1 2 Quadratic 3 Trinomial
4x3 3 Cubic 1 Monomial
9x4 + 11x 4 Fourth degree 2 Binomial
5 0 Constant 1 monomial
Once you simplify a polynomial by
combining like terms, you can name the
polynomial based on degree or number of
monomials it contains.
15. Classifying Polynomials
Write each polynomial in standard form. Then name each
polynomial based on its degree and the number of terms.
a) 5 – 2x
-2x + 5 Place terms in order.
linear binomial
b) 3x4 – 4 + 2x2 + 5x4 Place terms in order.
3x4 + 5x4 + 2x2 – 4 Combine like terms.
8x4 + 2x2 – 4
4th degree trinomial
16. Write each polynomial in standard
form. Then name each polynomial
based on its degree and the
number of terms.
a) 6x2 + 7 – 9x4
b) 3y – 4 – y3
c) 8 + 7v – 11v
and Subtracting
Subtracting
Polynomials
Polynomials
The sum or difference
18. Just as you can perform operations on
integers, you can perform operations on
polynomials. You can add polynomials using two
methods. Which one will you choose?
Closure of polynomials under addition or subtraction
The sum of two polynomials is a polynomial.
The difference of two polynomials is a polynomial.
Polynomials You can rewrite each polynomial,
inserting a zero placeholder for
the “missing” term.
Method 1 (vertically)
Line up like terms. Then add the coefficients.
4x2 + 6x + 7 -2x3 + 2x2 – 5x + 3
2x2 – 9x + 1 0 + 5x2 + 4x - 5
6x2 – 3x + 8 -2x3 + 7x2 – x - 2
Method 2 (horizontally)
Group like terms. Then add the coefficients.
(4x2 + 6x + 7) + (2x2 – 9x + 1) = (4x2 + 2x2) + (6x – 9x) + (7 + 1)
= 6x2 – 3x + 8
Example 2:
(-2x3 + 0) + (2x2 + 5x2) + (-5x + 4x) + (3 – 5)
Example 2
Use a zero placeholder
20. Simplify each sum
• (12m2 + 4) + (8m2 + 5)
• (t2 – 6) + (3t2 + 11) Remember
Use a zero as a placeholder
for the “missing” term.
• (9w3 + 8w2) + (7w3 + 4)
• (2p3 + 6p2 + 10p) + (9p3 + 11p2 + 3p )
Word Problem
21. Find the perimeter of
each figure
9c - 10 - 6
17x
8x - 2
5c + 2
5x
9x
+
1
Recall that the
perimeter of a figure is
the sum of all the sides.
22. Subtracting
Polynomials
Earlier you learned that subtraction means to add the
opposite. So when you subtract a polynomial, change
the signs of each of the terms to its opposite. Then
Method 1 (vertically)
Line up like terms. Change the signs of the second polynomial,
then add. Simplify (2x3 + 5x2 – 3x) – (x3 – 8x2 + 11)
2x3 + 5x2 – 3x 2x3 + 5x2 – 3x
-(x3 – 8x2 + 0 + 11) -x3 + 8x2 + 0 - 11
x3 +13x2 – 3x - 11
Remember,
the opposite.
Method 2
23. Method 2 (horizontally)
Simplify (2x3 + 5x2 – 3x) – (x3 – 8x2 + 11)
Write the opposite of each term.
2x3 + 5x2 – 3x – x3 + 8x2 – 11
Group like terms.
(2x3 – x3) + (5x2 + 8x2) + (3x + 0) + (-11 + 0) =
x3 + 13x2 + 3x - 11 =
x3 + 13x2 + 3x - 11
24.
25. Simplify each
subtraction
• (17n4 + 2n3) – (10n4 + n3)
• (24x5 + 12x) – (9x5 + 11x)
• 6c – 5 2b + 6 7h 2 + 4h - 8
-(4c + 9) -(b + 5) -(3h2 – 2h + 10)
26. Multiplying and
and Factoring
Factoring
Using the Distributive Property
27. Observe the rectangle below. Remember that
the area A of a rectangle with length l and
width w is A = lw. So the area of this
rectangle is (4x)(2x), as shown.
4x
2x
A = lw
A = (4x)(2x)
****************************
x+x+x+x
x
+
x
The rectangle above shows the example that
4x = x + x + x + x and 2x = x + x NEXT
28. We can further divide the rectangle into
squares with side lengths of x.
x +x+x+ x
x
+
x
Since each side of the squares
are x, then x · x = x2
x+x+x+x
x x2 x2 x2 x2
+
x x2 x2 x2 x2
By applying the area formula
for a rectangle, the area of the
rectangle must be (4x)(2x).
This geometric model suggests the following
algebraic method for simplifying the product
of (4x)(2x).
(4x)(2x) = (4 · x)(2 · x) = (4 · 2)(x · x) = 8x 2
NEXT
Commutative Property Associative Property
29. To simplify a product of monomials
(4x)(2x)
• Use the Commutative and Associative Properties
of Multiplication to group the numerical
coefficients and to group like variable;
(4x)(2x) = (4 · 2)(x · x ) =
• Calculate the product of the numerical
coefficients; and
(4 · 2) = 8
• Use the properties of exponents to simplify the
variable product.
(x · x) = x1 · x1 = x1+1 = x2
Therefore (4x)(2x) = 8x2
30. You can also use the Distributive Property for
multiplying powers with the same base when
multiplying a polynomial by a monomial.
Simplify -4y2(5y4 – 3y2 + 2) Remember,
Multiply powers with the same base:
35 · 34 = 35 + 4 = 39
-4y2(5y4 – 3y2 + 2) =
-4y2(5y4) – 4y2(-3y2) – 4y2(2) = Use the Distributive Property
-20y2 + 4 + 12y2 + 2 – 8y2 = Multiply the coefficients and add the
-20y6 + 12y4 – 8y2 exponents of powers with the same base.
31. Simplify each product.
a) 4b(5b2 + b + 6)
b) -7h(3h2 – 8h – 1)
c) 2x(x2 – 6x + 5)
Remember,
Multiplying powers with the same base.
d) 4y2(9y3 + 8y2 – 11)
32. Factoring a Monomial
from a Polynomial Factoring a polynomial
reverses the
multiplication process.
Find the GCF of the terms of: To factor a monomial
4x3 + 12x2 – 8x from a polynomial, first
List the prime factors of each term. find the greatest
4x3 = 2 · 2 · x · x x common factor (GCF) of
12x2 = 2 · 2 · 3 · x · x its terms.
8x = 2 · 2 · 2 · x
The GCF is 2 · 2 · x or 4x.
33. Find the GCF of the terms of each polynomial.
a) 5v5 + 10v3
b) 3t2 – 18
c) 4b3 – 2b2 – 6b
d) 2x4 + 10x2 – 6x
34. Factoring Out a To factor a polynomial
Monomial completely, you must factor
until there are no common
factors other than 1.
Factor 3x3 – 12x2 + 15x
Step 1
Find the GCF
3x3 = 3 · x · x · x Step 2
12x2 = 2 · 2 · 3 · x · x Factor out the GCF
15x = 3 · 5 · x 3x3 – 12x2 + 15x
The GCF is 3 · x or 3x = 3x(x2) + 3x(-4x) + 3x(5)
= 3x(x2 – 4x + 5)
35. Use the GCF to factor each polynomial.
a) 8x2 – 12x
b) 5d3 + 10d
c) 6m3 – 12m2 – 24m
d) 4x3 – 8x2 + 12x
Try to factor mentally by
scanning the coefficients of
each term to find the GCF.
Next, scan for the least power
of the variable.
36. Multiplying Binomials
Binomials
Using the infamous FOIL method
37. Using the
Distributive
Property Distribute x + 4
As with the other Simplify: (2x + 3)(x + 4)
examples we have
seen, we can also (2x + 3)(x + 4) =
use the Distributive
Property to find the 2x(x + 4) + 3(x + 4) =
product of two 2x2 + 8x + 3x + 12 =
binomials. 2x2 + 11x + 12
Now Distribute 2x and 3
38. Simplify each
product.
a) (6h – 7)(2h + 3)
b) (5m + 2)(8m – 1)
c) (9a – 8)(7a + 4)
d) (2y – 3)(y + 2)
39. Multiplying using FOIL
Another way to organize multiplying two binomials is
to use FOIL, which stands for,
“First, Outer, Inner, Last”. The term FOIL is a
memory device for applying the Distributive
Property to the product of two binomials.
Simplify (3x – 5)(2x + 7)
First Outer Inner Last
= (3x)(2x) + (3x)(7) – (5)(2x) – (5)(7)
(3x – 5)(2x + 7) = 6x2 + 21x - 10x - 35
= 6x2 + 11x - 35
The product is 6x2 + 11x - 35
40. Simplify each product
using FOIL
Remember,
First, Outer, Inner, Last
a) (3x + 4)(2x + 5)
b) (3x – 4)(2x + 5)
c) (3x + 4)(2x – 5)
d) (3x – 4)(2x – 5)
41. Applying
Multiplication of area of outer rectangle =
Polynomials. (2x + 5)(3x + 1)
area of orange rectangle =
Find the area of the x(x + 2)
Simplify. = area of outer rectangle – area of
2x + 5 orange portion
(2x + 5)(3x + 1) – x(x + 2) =
x+2
6x2 + 15x + 2x + 5 – x2 – 2x =
3x + 1
x
6x2 – x2 + 15x + 2x – 2x + 5 =
5x2 + 17x + 5
Use the Distributive Property
Use the FOIL method to
to simplify –x(x + 2)
simplify (2x + 5)(3x + 1)
42. Find the area of the
Simplify.
Find the area of the green shaded region. Simplify.
5x + 8
6x + 2 5x
x+6
43. FOIL works when you are multiplying two binomials.
However, it does not work when multiplying a trinomial
and a binomial.
(You can use the vertical or horizontal method to distribute each term.)
Remember multiplying
Simplify (4x + x – 6)(2x – 3)
2 whole numbers.
312
Method 1 (vertical) x 23
936
4x2 + x - 6
624
7176
2x - 3
-12x2 - 3x + 18 Multiply by -3
8x3 + 2x2 - 12x Multiply by 2x
8x3 - 10x2 - 15x + 18 Add like terms
44. Multiply using the
horizontal method.
Drawing arrows
Method 2 (horizontal) between terms can
products.
(2x – 3)(4x2 + x – 6)
= 2x(4x2) + 2x(x) + 2x(-6) – 3(4x2) – 3(x) – 3(-6)
= 8x3 + 2x2 – 12x – 12x2 – 3x + 18
= 8x3 -10x2 - 15x + 18
The product is 8x3 – 10x2 – 15x + 18
45. Simplify using the Distributive Property.
a) (x + 2)(x + 5)
b) (2y + 5)(y – 3)
c) (h + 3)(h + 4)
Simplify using FOIL.
a) (r + 6)(r – 4)
b) (y + 4)(5y – 8)
c) (x – 7)(x + 9)
WORD PROBLEM
46. Find the area of the