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Transcript
```Lesson 4
MA 15200
A Monomial or Term is a number or the product of a number and one or more variables with
whole-number exponents. The number factor is called a Coefficient of the term. The Degree of
the monomial (or term) is the sum of the exponents of it variables. A Constant has degree 0.
Ex 1: For each term, state the coefficient and the degree of the term.
a)
 14 x 3 y
b)
c)
d)
a5

2
r 2h
 5.1
A monomial or a sum of monomials is called a Polynomial. A polynomial with 2 terms is called
a Binomial, and a polynomial with 3 terms is called a Trinomial. The Degree of the
Polynomial is the degree of the term of the polynomial of the highest degree.
Ex 2: For each polynomial, state the number of terms and the degree.
a) 4 x3  5 x  9 x 4  3x 2
b)
12 x 2 y  7 x 2 y 2  8 xy 2
The leading coefficient of a polynomial is the coefficient of the term with the highest degree.
That term is also called the leading term.
Ex 3: State the degree of each polynomial. Identify any binomials or trinomials.
a)
4 x3  3x 2 y 2  3xy  4 y 2
b)
3
4
 ab 2c  a 2b 2c
4
3
Ex 4: State the leading term and the leading coefficient of this polynomial.
 4 xy  3x 2 y  7 x3 y  8 x
1
Monomials or Terms that have the same variable factors are called like terms and can be
combined by combining the coefficients and keeping the same variable factors.
1. Remove parentheses using the distributive property.
2. Combine any like terms.
a ) (3a 3b  7 a 2b  4ab)  (5a 2b  14a 3b  8ab) 
b)
(4 x3  4 x  2 x 2  6)  (2 x 2  9  3 x  x 3 ) 
c)
4(2m3  7 m 2 )  6(8m 2  m3 )  2(2m3  3m 2 ) 
Multiplication of Polynomials:
1.
To multiply two monomials:
Use regular 'rules of exponents'. Multiply the coefficients and multiply the variables.
2.
To multiply a monomial and a polynomial (binomial or more terms):
Use the distributive property. Multiply each term of the polynomial by the
monomial.
3.
To multiply two binomials:
Use FOIL (first, outer, inner, last)
4.
To multiply two general polynomials with 2 or more terms:
Use a variation of FOIL. Distribute each term of the first polynomial to every term of
the second polynomial and combine the results.
Ex 6: Multiply and write product in simplified form.

1
2

a)
(4a 2b)  ab 4 
c)
( m  2)(3m  5) 
e)
(4 x  3)( x 2  3 x  4) 
4 xy 2 (3x 3  2 x 2 y  xy 2 ) 
b)
d)
( 2  x)( 5  2 x) 
2
*Note: Only two polynomials (other than monomials) can be multiplied at one time.
To find the following product, multiply two of the binomials, then multiply that product by the
third binomial.
( x  3)( x  4)(2 x  1)
Special Products:
1. ( x  y )( x  y )  x 2  y 2
2.
These types of binomials are called conjugates .
( x  y ) 2  x 2  2 xy  y 2
(1st) 2  2(1st)(2nd)  (2nd) 2
3.
( x  y ) 2  x 2  2 xy  y 2
(1st) 2  2(1st)(2nd)  (2nd) 2
Ex 7: Find each product in simplest form.
a ) (4m  3n) 2 
b)
(6  5w)(6  5w) 
c)
(2 x  3) 2
d)
(4 y  5)(4 y  5)
e)
(2  x )(2  x )
3
The conjugate of a  b is a  b and the conjugate of a  b is a  b . The product of conjugates
with any square root terms will always be rational (without radicals).
To rationalize a denominator with a binomial, multiply numerator and denominator by the
conjugate of the denominator.
Ex 1: Rationalize the denominator and simplify where possible.
9
a)

6 3
b)
12 x

x 1
c)
4 5

3 3
4
```
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