Download Multiplying Polynomials

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I. Model Problems.
II. Practice
III. Challenge Problems
VI. Answer Key
Web Resources
Multiplying Polynomials
www.mathwarehouse.com/algebra/polynomial/how-to-multiply-polynomials.php
Multiplying a Polynomial by a Monomial
www.mathwarehouse.com/algebra/polynomial/multiply-polynomial-by-monomial.php
How to Multiply Binomials (FOIL)
www.mathwarehouse.com/algebra/polynomial/foil-method-binomials.php
Multiply Polynomials by Polynomials
www.mathwarehouse.com/algebra/polynomial/multiplying-polynomials-bypolynomials.php
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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 a monomial by a polynomial, use the distributive property.
Distributive Property
For real numbers a, b and c, a(b + c) = ab + ac
Example 1 Simplify 2x(3 + 5y).
= 2x(3) + 2x(5y)
= 6x + 10xy
Distributive property.
Simplify.
The answer is 6x + 10xy.
A binomial is a polynomial with two terms. To multiply two
binomials, use the FOIL method. FOIL stands for First, Inner, Outer,
Last. To use this method, calculate the products of the first, inner, outer
and last terms; then add. This is shown in the example.
Example 2 Simplify (x + 3)(y + 10).
Use the FOIL method as shown:
The product of the First terms is xy.
The product of the Inner terms is 3y.
The product of the Outer terms is 10x.
The product of the Last terms is 30.
The sum of the first, inner, outer and last terms is:
xy + 3y + 10x + 30.
The answer is xy + 3y + 10x + 30.
Larger polynomials can be multiplied together by repeatedly using the
Distributive property.
Example 3 Simplify (x + 2)(x2 + 3x + 5).
= x(x2) + x(3x) + x(5) + 2(x2) +
2(3x) + 2(5)
= x3 + 3x2 + 5x + 2x2 +6x + 10
Distributive property.
= x3 + 5x2 + 11x + 10
Combine like terms.
Multiply.
The answer is x3 + 5x2 + 11x + 10.
II. Practice
Simplify.
1. 5(x + 2)
2. 7x(x + y)
3. 5x(6x2 + 10)
4. 10y(6xy – 14)
5. –2x2(–2 + 6xy)
6. –6x(8 – 6x3)
7. (x + 2)(x + 5)
8. (x – 6)(x + 9)
9. (x + 3)(x + 3)
10. (x – 5)(x + 5)
11. (x + 5)(y + 10)
12. (x + y)(z + y)
13. (100 + x)(x – 100)
14. (x2 + 1)(x + 9)
15. (x – 5)(2x + 13)
16. (3m + 6)(4m + 11)
17. (11s + r)(–7r + s)
18. (6 – x)(x + 100)
19. (3x + 10)(2x – 5)
20. (–y + 2)(–x – 3)
21. (x + 2)(x2 + 5x + 6)
22. (y + 3)(y2 – 6y + 1)
23. (x + 3)(x2 + 7x + 11)
24. (y + 5)(y2 – 7y – 10)
25. (2x + 1)(x2 – 6x + 2)
26. (3y + 3)(4y2 + 5y + 20)
27. (3x + 5)(5x2 + 4x + 11)
28. (2y + 7)(8y2 – 6y + 1)
29. (x2 – 2x + 1)(x2 + 5x + 6)
30. (x2 – 6x + 2)(x2 + 3x + 2)
III. Challenge Problems
31. What is the area of a rectangle with length (3x + 2) inches and
width (4x + 10) inches? Write your answer as an expression in terms of
x.
32. Explain how using the FOIL method to multiply binomials is
similar to using the Distributive property.
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33. Correct the Error
There is an error in the student work shown below:
Question: Simplify (x – 3)(x – 7).
Solution:
(x – 3)(x – 7)
= x•x – 7•x – 3•x – 21
= x2 – 10x – 21
What is the error? Explain how to solve the problem.
_________________________________________________________
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IV. Answer Key
1. 5x + 10
2. 7x2 + 7xy
3. 30x3 + 50x
4. 60xy2 – 140y
5. 4x2 – 12x3y
6. –48x + 36x4
7. x2 + 7x + 10
8. x2 – 3x – 54
9. x2 + 6x + 9
10. x2 – 25
11. xy + 10x + 5y + 50
12. xz + xy + yz + y2
13. x2 – 10,000
14. x3 + 9x2 + x + 9
15. 2x2 + 3x – 65
16. 12m2 + 57m + 66
17. 11s2 – 76rs – 7r2
18. –x2 – 94x + 600
19. 6x2 + 5x – 50
20. yx + 3y – 2x – 6
21. x3 + 5x2 + 6x + 2x2 + 10x + 12 = x3 + 7x2 + 16x + 12
22. y3 – 6y2 + y + 3y2 – 18y + 3 = y3 – 3y2 – 17y + 3
23. x3 + 7x2 + 11x + 3x2 + 21x + 33 = x3 + 10x2 + 32x + 33
24. y3 – 7y2 – 10y + 5y2 – 35y – 50 = y3 – 2y2 – 45y – 50
25. 2x3 – 12x2 + 4x + x2 – 6x + 2 = 2x3 – 11x2 – 2x + 2
26. 12y3 + 15y2 + 60y + 12y2 + 15y + 60 = 12y3 + 27y2 + 75y + 60
27. 15x3 + 12x2 + 33x + 25x2 + 20x + 55 = 15x3 + 37x2 + 53x + 55
28. 16y3 – 12y2 + 2y + 56y2 – 42y + 7 = 16y3 + 44y2 – 40y + 7
29. x4 + 3x2 – 3x2 – 7x + 6
30. x4 – 3x3 – 14x2 – 6x + 4
31. 12x2 + 38x + 20 square inches
32. Multiplying to obtain the first and outer terms is the same as using
the distributive property; the first term of the left-hand binomial is
distributed to the right-hand binomial. Likewise, multiplying to obtain
the inner and last terms is equivalent to distributing the second term of
the left-hand binomial to the right-hand binomial.
33. The student wrote (-3)(-7) = -21 instead of +21.