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How to Multiply a Monomial by a Polynomial

Multiplying monomials by polynomials in algebra can be tricky, especially when it comes to keeping negative and positive signs straight. Here are some guidelines to getting it right. The key is to do signs, then numbers, then variables.

1. Step 1
Write the monomial to the left and the polynomial in parenthesis to the right.

2. Step 2
Think about each sign before you think about multiplying the numbers. If the monomial is positive and the first term (chunk) of the polynomial is negative think "positive X negative = positive" and write the correct sign down in your answer. Write a negative sign for a negative or a plus sign for a positive (expect for the very first chunk of your answer where no sign is needed if it is positive).

3. Step 3
Once you have determined the first sign, multiply the number in the monomial and the number in the first term (chunk) of the polynomial. Write down the answer after the sign.

4. Step 4
The third step is to use your exponent (power) rule to multiply any variables such as "X" "Y" etc. If you have X^3 times X^2 you will get X^5. Add the exponents and write down the variable with its new exponent(power) to the right of the sign and the number. Do this for each exponent that is in that chunk of the polynomial. If there is a variable that is only in the monomial or only in the polynomial, copy it down just like it is with the exponent(power) it has.

5. Step 5
Go through all three steps (1. sign, 2. multiply numbers, 3. use exponent rules on powers) for each chunk of the polynomial. This is the distributive property at work.