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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.
Addition/Subtraction of Polynomials:
To add or subtract polynomials:
1. Remove parentheses using the distributive property.
2. Combine any like terms.
Ex 5: Add/Subtract these polynomials
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