I. Model Problems
A monomial is an expression that is a number, variable or product of a
number and variables.
Examples of monomials: –3, 4x, 5xy, y2
A polynomial is a monomial or the sum or difference of monomials.
Examples of polynomials: 2x + 4, –x4 + 4x3 – 5x2, 400
To multiply monomials by a polynomial, multiply the monomial by
each of the terms of the polynomial.
Example Simplify 3x2(2x2 + 5x + 10).
= 3x2(2x2) + 3x2(5x) + 3x2(10)
Distributive property.
= 6x4 + 15x3 + 30x2
Multiply.
The answer is 6x4 + 15x3 + 30x2.
II. Practice
Simplify.
1. 3(x2 + 2x + 6)
2. 5(x2 + 4x – 3)
3. 2(x2 – x + 11)
4. 4(x2 + 2x + 30)
5. –3x(x2 – 7x + 1)
6. –6x(3x2 + 5x – 10)
7. 5x(–2x2 + 8x + 12)
8. 8x(2x2 + 6x + 10)
9. 3x2(2x2 + 20x + 12)
10. 6x2(–2x2 + 5x – 30)
11. 9x3(x2 + x + 1)
12. –5x6(3x2 – 12x + 30)
13. –2x3(3x2 + 4x + 5)
14. x2y2(x2 + 2x + 1)
15. 4x2y(–x2 – x + 10)
16. 5x2y(x2 + 2xy + 7)
17. 3xy(2x2 + 2xy + y2)
18. 4xy2(3x2 – 5xy + 6y2)
19. 3x2y2z(7x2 + 2y + z + 4w)
20. –2xyz(3x2 + 2y2 + z2 – 4w2)
III. Challenge Problems
21. What is the area of a rectangle with length 3xy inches and width
(3x2 + 6x + 10) inches? Write your answer as an expression in terms of
x and y.
22. The diagram shows a soccer field divided into three sections. The
measurements shown are in feet. Find the area of the soccer field.
23. Correct the Error
There is an error in the student work shown below:
Question: Simplify x2(3x2 + 5y + 6).
Solution:
x2(3x2 + 5y + 6)
= x2•(3x2) + x2•5y + x2•6
= 3x4 + 5x3 + 6x2
What is the error? Explain how to solve the problem.
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IV. Answer Key
1. 3x2 + 6x + 18
2. 5x2 + 20x – 15
3. 2x2 – 2x + 11
4. 4x2 + 8x + 120
5. –3x3 + 21x2 – 3x
6. –18x3 – 30x2 + 60x
7. –10x3 + 40x2 + 60x
8. 16x3 + 48x2 + 80x
9. 6x4 + 60x3 + 36x2
10. –12x4 + 30x3 – 18x2
11. 9x5 + 9x4 + 9x3
12. –15x8 + 60x – 150x6
13. –6x5 – 8x4 – 10x3
14. x4y2 + 2x3y2 + x2y2
15. –4x4y – 4x3y + 40x2y
16. 5x4y + 10x3y2 + 35x2y
17. 6x3y + 6x2y2 + 3xy3
18. 12x3y2 – 20x2y3 + 24xy4
19. 21x4y2z + 6x2y3z + 3x2y2z2 + 12x2y2zw
20. –6x3yz – 4xy2z – 2xyz3 + 8xyzw2
21. 9x3y + 18x2y + 30xy in.2
22. 12x4y + 8x3y + 20x2y square feet
23. The student incorrectly multiplied x2(5y). The answer should be 3x4
+ 5x2y + 6x2.