Download Adding, Subtracting, and Multiplying Polynomials Monomial: An

Adding, Subtracting, and Multiplying Polynomials
Monomial: An expression that is a number, a variable, or numbers and variables multiplied together.
Monomials only have variables with whole number exponents and never have variables in the
denominator of a fraction or variables under roots.
xyz
1
, − w, 23, x 2 , x3 y 4
Monomials: 5b,
8
3
1
Not Monomials: 4 ,
x
3
−1
x, a , z
1
5
Constant: A monomial that contains no variables, like 23 or −1.
Coefficient: The numerical part of a monomial (the number being multiplied by the variables.)
Polynomial: A monomial or several monomials joined by + or – signs.
Terms: The monomials that make up a polynomial. Terms are separated by + or − signs.
Like Terms: Terms whose variables and exponents are exactly the same.
Binomial: A polynomial with two unlike terms.
Trinomial: A polynomial with three unlike terms.
Examples: Decide whether each expression is a polynomial. If it isn’t, explain why not.
4
12
a) 5 x 4 + 2 x 3 + 6 x
b) − a − a 5
c)
d) 6c −2 + c − 1
3
m+2
1
e) 6 z 2 + 5 z 2 − 2
f) 7
g) −8n − 3
h) 3 x + 2
Adding and Subtracting Polynomials
To add or subtract polynomials, combine like terms. Add or subtract the coefficients. The variables and
exponents do not change. Remember to subtract everything inside the parentheses after a minus sign.
Subtract means “add the opposite,” so change the minus sign to a plus sign and then change the signs of
all the terms inside the parentheses.
Examples: Simplify each expression.
a) ( 5n 2 − 2 ) + ( 7 − 3n 2 )
b) ( 2r 2 + 5r ) + ( r 2 − 4r )
c) ( 4 x 2 − 3 x + 1) + ( −2 x 2 + 5 x − 6 )
d) ( 7 z 2 + 12 z − 5 ) + ( 6 z − 4 z 2 − 3)
e) ( 2 w2 + 3w ) − ( 4 w2 + w )
f) ( u 3 − 4u 2 + u ) − ( 2u 2 − 5u 3 )
g) ( −6 x 2 − 3 x + 2 ) − ( −4 x 2 − x + 3)
h) ( 4 y 2 + 12 y − 7 ) − ( 20 y + 5 y 2 − 8 )
i) ( 6m 2 + 5m ) − ( 4m 2 − 2m ) + ( 3m 2 − 7 m )
j) ( −2k + 5 ) + ( k 2 − 3k ) − ( −4k 2 + 8 )
Multiplying Polynomials
To multiply two polynomials, multiply each term of one polynomial by each term of the other
polynomial. Then combine any like terms. When you are multiplying two binomials, this is sometimes
called the FOIL Method because you multiply F the first terms, O the outside terms, I the inside terms,
and L the last terms.
Examples: Multiply.
a) − xy ( 7 x 2 y + 3 xy − 11)
b) ( m + 3)( m − 8 )
c) ( 3 x + 1)( 5 x − 2 )
d) ( −2 z + 5 )( −5 z − 8)
e) ( t 2 − 4 ) ( 2t + 9 )
f) ( 2u 2 − 1)( −5u 2 + 4 )
g) ( z + 5 )
2
i) ( n + 3)( n − 3)
k)
( 2 x − 3) ( 5 x 2 − 6 x + 7 )
h) ( 2 x − 3)
2
j) ( 5 y − 2 )( 5 y + 2 )
l)
( 4x
2
+ 7 x − 3)( x 2 − 2 x + 8 )