Unit 3a Multiplying Polynomials Assignment
( /40)
4. Demonstrate an understanding of the multiplication of polynomial expressions (limited to
monomials, binomials and trinomials), concretely, pictorially and symbolically.
1. Combine like terms to simplify ( /4)
a) 6𝑥 − 3𝑦 + 2𝑥 + 𝑦 − 2𝑥 + 4𝑦
b) 2𝑔 − 5(𝑔 + 8ℎ) − 7ℎ
c) 5 − 3 + 9(𝑞 − 7)
d) 3𝑥𝑦 + 5𝑦 − 4𝑥 + 7𝑦 − 4𝑥𝑦 + 2𝑥 + 5𝑦𝑥
4.1 Model the multiplication of two given binomials, concretely or pictorially, and record the process
symbolically.
4.2 Relate the multiplication of two binomial expressions to an area model.
2. Simplify (𝟒𝒙 − 𝟏)(𝟐𝒙 − 𝟑) by using… ( /3)
a. An area diagram
b) algebra tiles
property
c) distributive
4.2 Relate the multiplication of two binomial expressions to an area model.
3. Multiply the following polynomials using the distributive property. ( /4)
a. −2𝑥(5𝑥 − 2𝑦)
b. 2𝑥(𝑥 2 − 4𝑥 + 3)
c. (𝑥 − 2𝑦)(4𝑥 − 5𝑦)
d. (2𝑝 + 4)(2𝑝 − 4)
4.5 Multiply two polynomials symbolically, and combine like terms in the product.
4. Use an area diagram to simplify. ( /4)
a. −7𝑎(4𝑎 + 5𝑏 − 3)
c.
(2𝑥 2 − 3𝑥 + 2)(−2𝑥 2 + 3𝑥 − 4)
b. (𝑥 2 + 3𝑥 + 4)(𝑥 2 + 𝑥)
d. (5𝑗 − 6)2
4.1 Model the multiplication of two given binomials, concretely or pictorially, and record the process
symbolically.
5. Use algebra tiles to simplify the following polynomials ( /2)
a. −2𝑘(𝑘 + 4)
b. (2𝑘 − 3)(−𝑘 − 2)
4.6 Generalize and explain a strategy for multiplication of polynomials.
6. Use your preferred method to expand and simplify the following expressions.
( /9)
a. (2𝑛 + 4)(3𝑛 + 2)(2𝑛 − 1)
b. (4𝑤 + 5)(2𝑤 − 3)
c. (9 + 𝑚)3
d. 4𝑡(𝑡 − 8) + (𝑡 − 6)(5𝑡 + 3)
𝑒. 2(3𝑡 − 5)2
4.3 Explain, using examples, the relationship between the multiplication of binomials and the
multiplication of two-digit numbers.
7. Use an area diagram to find the product of 24 and 43. ( /2)
4.4 Verify a polynomial product by substituting numbers for the variables.
4.6 Generalize and explain a strategy for multiplication of polynomials.
8. Word Problems
a. Bob trimmed a square photo to fit into a rectangular frame. He cut 8
cm from one side of the photo and 2 cm from the other side. ( /3)
i) Write an expression for the area and perimeter of the trimmed photo.
ii) Simplify the area expression found in i)
b. The length of a rectangular prism is 2𝒄 − 𝟐. The height of the prism is
𝒄 + 𝟕 with a width of 3𝒄 − 𝟒. ( /3)
i) Write an expression for the volume of the prism.
ii)
iii)
Expand the expression you found in i)
Verify your expansion by substituting x = 2 into each polynomial.
c. Use the diagram to the right to answer the following questions. ( /3)
i)
Develop an expression to represent the area of the polygon.
ii)
Expand and simplify the expression found above
iv)
Verify your expansion by substituting x = 4 into each polynomial.
d. A rectangle has length 4x+3 cm and width of 2x+4 cm. A square
inside it has a side length of x+3 cm. ( /3)
i)
Write an expression to represent the area of the shaded region.
ii)
Expand and simplify the expression in i).
iii)
Verify your expansion by substituting x = 4 into each polynomial.
Unit 3b Factoring Polynomials Assignment ( /40)
5. Demonstrate an understanding of common factors and trinomial factoring, concretely, pictorially and
symbolically.
5.1 Determine the common factors in the terms of a polynomial, and express the polynomial in factored
form.
1. Factor the following polynomials by removing a gcf. [4 marks]
a) 4𝑝𝑞𝑟 − 2𝑝𝑞𝑠 + 8𝑝𝑞𝑡
b) 8𝑥 + 2𝑥 2
c) 5𝑥𝑦 2 − 10𝑤𝑥 2 𝑦
d) 8𝑎2 𝑏 3 𝑐 4 + 4𝑐 5 𝑎4 𝑏 3 − 10𝑏 4 𝑎5 𝑐 2
5.3 Factor a polynomial that is a difference of squares, and explain why it is a special case of trinomial
factoring where b =0.
2. How is it possible to factor a 2 term difference of squares question using the diamond
method? Explain your answer using the example 𝑥 2 − 25. [2 marks]
3. Factoring the following binomials if possible. [8 marks]
a) 4𝑥 2 − 49
e) 25𝑥 2 − 100
b) 𝑦 2 − 𝑥 2
f) −4ℎ2 + 25𝑔2
c) 3𝑥 2 − 48
g) 𝑥 4 − 81
d) 𝑥 2 + 4
h) 3𝑥 4 − 3
5.1 Determine the common factors in the terms of a polynomial, and express the polynomial in factored
form.
5.5 Factor a polynomial, and verify by multiplying the factors.
4. Factor the following polynomials using the grouping method. [4 marks]
a) 𝑥 2 + 3𝑥 + 6𝑥 + 18
b) 2𝑥 2 + 4𝑥 + 6𝑥 + 12
c) 15 − 3𝑦 − 5𝑦 + 𝑦 2
d) 𝑥 3 + 2𝑥 2 + 𝑥 + 2
5. Pick one of the questions above and verify your answers. [1 mark]
5.1 Determine the common factors in the terms of a polynomial, and express the polynomial in factored form.
5.2 Model the factoring of a trinomial, concretely or pictorially, and record the process symbolically.
5.5 Factor a polynomial, and verify by multiplying the factors.
5.6 Explain, using examples, the relationship between multiplication and factoring of polynomials.
5.7 Generalize and explain strategies used to factor a trinomial.
6. Factor the following trinomials. [8 marks]
a) 𝑎2 + 8𝑎 + 12
e) 𝑒 2 − 3𝑒 − 28
b) 𝑚2 + 18𝑚 + 81
f) 𝑝2 − 12𝑝𝑞 + 11𝑞 2
c) 3𝑝2 − 23𝑝 − 8
g) 12𝑚2 − 11𝑚 − 5
d) 5𝑎2 − 20𝑎𝑏 + 15𝑏 2
h) 2𝑡 𝟑 + 2𝑡 2 − 40𝑡
7. Pick two questions above and verify your solution. [2 marks]
Name:
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5.4 Identify and explain errors in a polynomial factorization.
5.5 Factor a polynomial, and verify by multiplying the factors.
5.6 Explain, using examples, the relationship between multiplication and factoring of polynomials.
8. Identify and correct the errors made when the following polynomials were factored. [2 marks]
a) 4𝑥 2 − 32𝑥 + 48
b) 4x2 – 64
(4𝑥 − 8)(𝑥 − 6)
4 (𝑥 − 4)(𝑥 − 4)
5.7 Generalize and explain strategies used to factor a trinomial.
5.8 Express a polynomial as a product of its factors.
Word Problems
9. A rectangular garden has an area of 2𝑤 2 + 3𝑤 − 5 square meters. The length and width of the
garden are to be increased by 1 meter on each side. Determine the length and width of the new
garden. [2 marks]
10. The volume of water in a tank is 2𝑥 3 + 5𝑥 2 − 3𝑥 cubic meters. Determine the length, width,
and height of the tank. [2 marks]
Name:
Page 10
11. The surface area of a cylinder is given by the formula 𝐴 = 2𝜋𝑟 2 + 2𝜋𝑟ℎ, where r is the radius
of the base and h is the height of the cylinder. Write the formula for A in factored form.
[1 marks]
12. The floor of a room is rectangular with an area of 81𝑚2 − 4𝑛2 . Determine the length and
width of the room and then write a polynomial expression that would represent the perimeter of
the room. [2 marks]
13. A section is missing from the expression 8𝑥 2 ±? ? ? +3. Identify two different possibilities for
the middle term of the expression and then factor the trinomial. [2 marks]