Teaching Notes 5.7: Multiplying a Polynomial by a Monomial
Multiplying a polynomial by a monomial requires application of the distributive property.
Mistakes often occur in use of the distributive property, computation with integers, or application
of the property of exponents.- Discuss the distributive property,a(b+c)=ab+ac, with your students. Provide examples
such as 3(8+4)= 3 × 8 + 3 × 4 = 24 + 12 =36. Note that each term in the parentheses
is multiplied by 3. Also note that if students follow the order of operations by simplifying
the expression inside the parentheses first, 3×12 also equals 36. Depending on the ability
of your students, you may wish to review integer operations, particularly subtraction and
multiplication. - Ask your students to simplify 2a(4a^3 +6) and demonstrate the application of the distribu-
tive property. Because the expression inside the parentheses cannot be simplified, students
must use the distributive property. Emphasize that each term in the polynomial must be
multiplied by the monomial and remind your students that they must now use the property
of exponents for multiplication,xm·xn=xm+n.Thus,2a(4a^3 +6)=(2a)(4a^3 )+(2a)(6)=
8 a^4 + 12 a. Note that they cannot add 8a^4 and 12abecause they are not similar terms. - Review the information and example on the worksheet with your students. Be sure that your
students understand how to apply the distributive property and the property of exponents
for multiplication. Discuss the example, explaining the steps in detail.
EXTRA HELP:
Be careful when you multiply the polynomial by a monomial that has a negative coefficient. Apply
the rules for multiplying integers.ANSWER KEY:
(1) 5 x+ 10 (2) 3 x− 12 (3) 4 x^2 + 20 x (4) 2 x^5 − 14 x^2
(5)− 12 a^3 − 3 a (6) 30 a^6 − 35 a^2 (7) 24 x^4 y+ 32 x^3 y^2 (8) 7 x^5 + 35 x^3 − 14 x^2
(9) 6 a^3 b^2 + 24 a^5 b^3 + 48 a^2 b^8
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(Challenge)Khalil is partially correct. His work is right but his answer is not in simplest form.
It should be 7x^2 − 14 x.
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