### Multiplication of a Polynomial by a Monomial

To multiply a polynomial by a monomial, use the distributive property: multiply each term of the polynomial by the monomial. This involves multiplying coefficients and adding exponents of the appropriate variables.

*Example 1*: 3*y*^{2}(12*y*^{3} -6*y*^{2} + 5*y* - 1) =?

= 3*y*^{2}(12*y*^{3}) + (3*y*^{2})(- 6*y*^{2}) + (3*y*^{2})(5*y*) + (3*y*^{2})(- 1)

= (3)(12)*y*^{2+3} + (3)(- 6)*y*^{2+2} + (3)(5)*y*^{2+1} + (3)(- 1)*y*^{2}

= 36*y*^{5} -18*y*^{4} +15*y*^{3} -3*y*^{2}

*Example 2*: -4*x*^{3}*y*(- 2*y*^{2} + *xy* - *x* + 9) =?

= - 4*x*^{3}*y*(- 2*y*^{2}) + (- 4*x*^{3}*y*)(*xy*) + (- 4*x*^{3}*y*)(- *x*) + (- 4*x*^{3}*y*)(9)

= (- 4)(- 2)*x*^{3}*y*^{1+2} + (- 4)*x*^{3+1}*y*^{1+1} + (- 4)(- 1)*x*^{3+1}*y* + (- 4)(9)*x*^{3}*y*

= 8*x*^{3}*y*^{3} -4*x*^{4}*y*^{2} +4*x*^{4}*y* - 36*x*^{3}*y*

### Multiplication of Binomials

To multiply a binomial by a binomial--(*a* + *b*)(*c* + *d* ), where *a*,
*b*, *c*, and *d* are terms--use the distributive property twice.
First, treat the second binomial as a single term and distribute over
the first binomial:

(a + b)(c + d )= a(c + d )+ b(c + d ) |

Next, use the distributive property over the second binomial:

a(c + d )+ b(c + d )= ac + ad + bc + bd |

At this point, there should be 4 terms in the answer -- every combination of a term of the first binomial and a term of the second binomial. Simplify the answer by combining like terms.

We can use the word **FOIL** to remember how to multiply two binomials (*a* + *b*)(*c* + *d* ):

- Multiply their
**F**irst terms. (*ac*) - Multiply their
**O**utside terms. (*ad*) - Multiply their
**I**inside terms. (*bc*) - Multiply their
**L**ast terms. (*bd*) - Finally, add the results together:
*ac*+*ad*+*bc*+*bd*. Combine like terms.

*Example 1.*(*xy* + 6)(*x* + 2*y*) =?

= (*xy*)(*x*) + (*xy*)(2*y*) + (6)(*x*) + (6)(2*y*)

= *x*^{2}*y* + 2*xy*^{2} + 6*x* + 12*y*

*Example 2.*(3*x*^{2} +7)(4 - *x*^{2}) =?

= (3*x*^{2})(4) + (3*x*^{2})(- *x*^{2}) + (7)(4) + (7)(- *x*^{2})

= 12*x*^{2} -3*x*^{4} +28 - 7*x*^{2}

= - 3*x*^{4} + (12 - 7)*x*^{2} + 28

= - 3*x*^{4} +5*x*^{2} + 28

*Example 3*: (*y* - *x*)(- 4*y* - 3*x*) =?

= (*y*)(- 4*y*) + (*y*)(- 3*x*) + (- *x*)(- 4*y*) + (- *x*)(- 3*x*)

= - 4*y*^{2} -3*xy* + 4*xy* + 3*x*^{2}

= 3*x*^{2} + (- 3 + 4)*xy* - 4*y*^{2}

= 3*x*^{2} + *xy* - 4*y*^{2}

### Multiplication of Polynomials

The strategy for multiplying two polynomials in general is similar to
multiplying two binomials. First, treat the second polynomial as a
single term, and distribute
over the first term:

(a + b + c)(d + e + f )= a(d + e + f )+ b(d + e + f )+ c(d + e + f ) |

Next, distribute over the second polynomial:

a(d + e + f )+ b(d + e + f )+ c(d + e + f )= ad + ae + af + bd + be + bf + cd + ce + cf |

At this point, the number of terms in the answer should be the number in the first polynomial times the number in the second polynomial--every combination of a term of the first polynomial and a term of the second polynomial. Since there are 3 terms in each polynomial in this example there should be 3(3) = 9 terms in our answer so far. If the first polynomial had 4 terms and the second had 5, there would be 4(5) = 20 terms in the answer so far.

Finally, since the the terms in such a product of polynomials are often highly redundant (many have the same variables and exponents), it is important to combine like terms.

*Example 1*: (*x*^{2} -2)(3*x*^{2} - 3*x* + 7) =?

= *x*^{2}(3*x*^{2} -3*x* + 7) - 2(3*x*^{2} - 3*x* + 7)

= *x*^{2}(3*x*^{2}) + *x*^{2}(- 3*x*) + *x*^{2}(7) - 2(3*x*^{2}) - 2(- 3*x*) - 2(7) (6 terms)

= 3*x*^{4} -3*x*^{3} +7*x*^{2} -6*x*^{2} + 6*x* - 14

= 3*x*^{4} -3*x*^{3} + (7 - 6)*x*^{2} + 6*x* - 14

= 3*x*^{4} -3*x*^{3} + *x*^{2} + 6*x* - 14

*Example 2*: (*x*^{2} + *x* + 3)(2*x*^{2} - 3*x* + 1) =?

= *x*^{2}(2*x*^{2} -3*x* + 1) + *x*(2*x*^{2} -3*x* + 1) + 3(2*x*^{2} - 3*x* + 1)

= *x*^{2}(2*x*^{2}) + *x*^{2}(- 3*x*) + *x*^{2}(1) + *x*(2*x*^{2}) + *x*(- 3*x*) + *x*(1) + 3(2*x*^{2}) + 3(- 3*x*) + 3(1) (9 terms)

= 2*x*^{4} -3*x*^{3} + *x*^{2} +2*x*^{3} -3*x*^{2} + *x* + 6*x*^{2} - 9*x* + 3

= 2*x*^{4} + (- 3 + 2)*x*^{3} + (1 - 3 + 6)*x*^{2} + (1 - 9)*x* + 3

= 2*x*^{4} - *x*^{3} +4*x*^{2} - 8*x* + 3

**Note:** To check your answer, pick a value for the variable and
evaluate both the original expression and your answer--they should
be the same.