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REVIEW
Like terms:
Like terms are term with exactly the same variables
raised to the exactly the same powers. Any constants in an
expression are considered like terms. Terms that are not
like terms are called unlike terms.
Like Terms
Unlike Terms
2x, 3x, -4x
2x, 2x2
Same variables, each
with a power of 1.
Different powers
3, 5, -1
3, 3x, 3x2
Constants
Different powers
5x2, -x2
5x2, 5y2
Same variables and
same powers
Different variables
Combining like terms:
Combining like terms is to add or subtract like terms.
Combine like terms containing variables by combining
their coefficients and keeping the same variables with the
same exponents.
Example: 3x2 – 8x2 = -5x2
Example: 5x + 2x = 7x
Example: (3x2 + 5x + 2) + (x2 – x - 3)
=3x2 + x2 + 5x-x + 2-3
= 4x2 + 4x - 1
POLYNOMIALS
A Polynomial is a term or sum of terms in which all variables
have whole number exponents.
Example: 2, 3x + 1, ½ x2 + 4x + 1,
Not polynomials: x -1, 3 x0.5 +2, x ¼ + 7x,
x
Monomial: Has one term (such as 5x2, or -6x, or 29)
Binomial: Has two unlike terms (such as 2x – 1, or x2 + 4x)
Trinomial: Has three unlike terms (such as ½ x2 + 4x + 1)
The degree of a polynomial is the highest exponent among all
the terms. If the polynomial is a constant, the degree is 0.
The polynomial 3x3 + 5
is a binomial with degree 3.
What type of polynomial is -2x2 + x - 3?
What degree is it?
Evaluate this polynomial for x = 3.
-2(3)2 + (3) – 3
= -2(9) + 3 - 3
= -18 + 3 – 3
= -15 -3
= -18
Example
The polynomial -16t2 + 28t + 8
gives the height (in feet) of an object t seconds after it has
been thrown straight upward. Find the height of the object
in 1 second.
The value of the polynomial when t=1 is the height of the
object after 1 second.
Height after 1 sec = -16(1)2 + 28(1) + 8
= -16 +28 +8
= 12 + 8
= 20 feet
Multiplying Polynomials
The distributive property is used when multiplying polynomials.
3x(x +5)
= 3x∙x + 3x ∙5
= 3x2 + 15x
Multiplying Binomials
(3x + 5)(2x – 1)
= (3x + 5)(2x) + (3x + 5)(-1)
= (3x)(2x) + (5)(2x) + (3x)(-1) + 5(-1)
= 6x2 + 10x + -3x + -5
Now combine like terms:
=6x2 + 7x - 5
A shortcut to this method when multiplying binomials is called
FOIL (First, Outer, Inner, Last)
(3x + 5)(2x – 1)
The First terms in each of these binomials are 3x and 2x.
The Outer terms are the ones on the “outsides” of the binomials, 3x and -1.
The Inner terms are the ones in the middle, 5 and 2x.
The Last terms are the second terms of each binomial, 5 and -1.
FOIL = (3x)(2x) + (3x)(-1) + (5)(2x) + 5(-1)
= 6x2 + -3x + 10x + -5
Now combine like terms:
=6x2 + 7x – 5
Also, there is the funny old man method….
left eyebrow right eyebrow
(3x + 5)(2x – 1)
nose
mouth
= (3x)(2x) + (5)(-1) + (5)(2x) + (3x)(-1)
= 6x2 -5 + 10x -3x = 6x2 – 5 + 7x
= 6x2 + 7x - 5
Special Products of Binomials
The Sum and Difference of Two Terms
(a + b)(a - b)=a2 – b2
FOIL:
a2 + ab – ab – b2
The inner terms and outer terms will always cancel each other out,
so you are left with:
a2 – b2
Examples:
(x + 2)(x – 2) = x2 +(2)(-2) = x2 – 4
(2x + 3)(2x -3) = (2x)2 +(3)(-3) = 4x2 – 9
The Square of a Binomial
(a + b)2 = a2 + 2ab + b2
Examples:
(x + 3)2 = x2 + 2(3)(x) + 32 = x2 + 6x + 9
(3x – 2)2 = (3x)2 + 2(3x)(-2) + (-2)2 = 9x2 -12x + 4
YOU TRY: (6x – y)2
Multiplying Larger Polynomials
The Vertical Method of multiplication is sometimes more efficient.
(3x + 2)(x2- x+4)
x2  x  4
 3x  2
2x2  2x  8
3 x 3  3 x 2  12 x
3 x 3  x 2  10 x  8
Multiply :
x
2

 2 x  7 x  2
x2  2x  7

x2
Just as you would in multiplying whole
numbers, put the larger polynomial (the one
with more terms) on top and the smaller
polynomial on the bottom.
Multiply every term in the larger polynomial
by 2 (the last term in the other polynomial).
Then on the next line multiply every term in
the larger polynomial by the next term in the
smaller polynomial (3x). Make sure to line up
like terms.
Rules for Variable Expressions:
Only like terms can be added, and when adding like terms, do not
change the exponent of the variable.
5x2 + 3x2 = 8x2
When multiplying variable expressions, add exponents of like variables
(5xy3)(2y2)=10xy3+2 = 10xy5
When taking powers of variable expression that is a monomial (one
term), multiply exponents of EVERY term inside the parentheses.
(2x3y4)3 = 23x3*3y4*3 = 8x9y12
When taking powers of a variable expression that is a binomial,
trinomial or some other polynomial, use the rules of polynomial
multiplication.
For example: (x+2)2 ≠ x2 + 22
(x+2)2 = (x+2)(x+2) = x2+4x + 4
(FOIL METHOD)
Example 2:
(x2+3x+5)2 = (x2+3x+5)(x2+3x+5) (DISTRIBUTIVE PROPERTY)
=(x2+3x+5)(x2)+ (x2+3x+5)(3x) + (x2+3x+5)(5)
=(x2)(x2) + (3x)(x2) + 5x2 + 3x(x2) + 3x(3x) + 3x(5) + 5x2+5(3x) + 5(5)