Multiplying a Polynomial by a Monomial
To multiply a monomial by a polynomial with more than one term,
use the Distributive Property to distribute the monomial to each term
in the polynomial.
Example 1Find: 8w^{3}y(4w^{2}y^{5}  w^{4})
Solution 

8w^{3}y(4w^{2}y^{5}  w^{4}) 
Multiply each term in
the polynomial by the
monomial, 8w^{3}y. 

= (8w^{3}y)(4w^{2}y^{5})  (8w^{3}y)(w^{4}) 
Within each term, write the coefficients next to each other. Write the
factors with base w next to each other and write the factors with base y
next to each other. 

= (8 Â· 4)(w^{3}
Â· w^{2})(y
Â· y^{5})  (8)(w^{3}
Â· w^{4})(y) 
Use the Multiplication
Property of Exponents. 

= (8 Â· 4)(w^{3 + 2} y^{1
+ 5})  (8)(w^{3 + 4 }y) 
Simplify. 

= 32w^{5}y^{6} + 8w^{7}y 
Example 2
Find: 5x^{4}(3x^{2}y^{2}  2xy^{2} + x^{3}y)
Solution 

5x4(3x^{2}y^{2}  2xy^{2} + x^{3}y)

Multiply each term in the polynomial by the monomial, 5x^{4}.


= (5x^{4})(3x^{2}y^{2})  (5x^{4})(2xy^{2})
+ (5x^{4})(x^{3}y) 
Within each term, write the coefficients next to each other. Write the
factors with base x next to each other and write the factors with base y next
to each other. 

= (5 Â· 3)(x^{4
}x^{ 2 }y^{2})  (5 Â·
2)(x^{4 }x^{ 1}y^{2}) + (5
Â· 1)(x^{4 }x^{ 3 }y)

Use the Multiplication Property of Exponents. 

= (5 Â· 3)(x^{4 + 2
}y^{2})  (5 Â· 2)(x^{4
+ 1} y^{2}) + (5 Â· 1)(x^{4
+ 3 }y) 
Simplify. 

= 15x^{6}y^{2}
 10x^{5}y^{2} + 5x^{7}y

