# SPECIAL PRODUCTS OF BINOMIALS WORKSHEET

Problems 1-10 : Expand.

Problem 1 :

(y + 3)2

Problem 2 :

(2p + 3q)2

Problem 3 :

(5 + k2)2

Problem 4 :

(-x + 2)2

Problem 5 :

(1 + z3)2

Problem 6 :

(m - 5)2

Problem 7 :

(5k - 1)2

Problem 8 :

(4e - 5f)2

Problem 9 :

(4 - d2)2

Problem 10 :

(p2 - q2)2

Problems 11-14 : Expand.

Problem 11 :

(m + n)(m - n)

Problem 12 :

(x + 3)(x - 3)

Problem 13 :

(c2 + 2d)(c2 - 2d)

Problem 14 :

(7 + y)(7 - y)

Problem 15 :

A square koi pond is surrounded by a gravel path. Write an expression that represents the area of the path.  Use the rule for (a + b)2.

(a + b)2 = a2 + 2ab + b2

Identify a and b : a = y and b = 3.

(y + 3)2 = y2 + 2(y)(3) + 32

= y2 + 6y + 9

Use the rule for (a + b)2.

(a + b)2 = a2 + 2ab + b2

Identify a and b : a = 2p and b = 3q.

(2p + 3q)2 = (2p)2 + 2(2p)(3q) + (3q)2

= 4p2 + 12pq + 9q2

Use the rule for (a + b)2.

(a + b)2 = a2 + 2ab + b2

Identify a and b : a = 5 and b = k2.

(5 + k2)2 = 52 + 2(5)(k2) + (k2)2

= 25 + 10k2 + k4

Use the rule for (a + b)2.

(a + b)2 = a2 + 2ab + b2

Identify a and b : a = -x and b = 2.

(-x + 2)2 = (-x)2 + 2(-x)(2) + 22

= x2 - 4x + 4

Use the rule for (a + b)2.

(a + b)2 = a2 + 2ab + b2

Identify a and b : a = 1 and b = z3.

(1 + z3)2 = 12 + 2(1)(z3) + (z3)2

= 1 + 2z3 + z6

Use the rule for (a - b)2.

(a - b)2 = a2 - 2ab + b2

Identify a and b : a = m and b = 5.

(m - 5)2 = m2 - 2(m)(5) + 52

= m2 - 10m + 25

Use the rule for (a - b)2.

(a - b)2 = a2 - 2ab + b2

Identify a and b : a = 5k and b = 1.

(5k - 1)2 = (5k)2 - 2(5k)(1) + 12

= 25k2 - 10k + 1

Use the rule for (a - b)2.

(a - b)2 = a2 - 2ab + b2

Identify a and b : a = 4e and b = 5f.

(4e - 5f)2 = (4e)2 - 2(4e)(5f) + (5f)2

= 16e2 - 40ef + 25f2

Use the rule for (a - b)2.

(a - b)2 = a2 - 2ab + b2

Identify a and b : a = 4 and b = d2.

(4 - d2)2 = 42 + 2(4)(d2) + (d2)2

= 16 + 8d2 + d4

Use the rule for (a - b)2.

(a - b)2 = a2 - 2ab + b2

Identify a and b : a = p2 and b = q2.

(p2 - q2)2 = (p2)2 + 2(p2)(q2) + (q2)2

= p4 + 2p2q2 + q4

Use the rule for (a + b)(a - b).

(a + b)(a - b) = a2 - b2

Identify a and b : a = m and b = n.

(m + n)(m - n) = m2 - n2

Use the rule for (a + b)(a - b).

(a + b)(a - b) = a2 - b2

Identify a and b : a = x and b = 3.

(x + 3)(x - 3) = x2 - 32

= x2 - 9

Use the rule for (a + b)(a - b).

(a + b)(a - b) = a2 - b2

Identify a and b : a = c2 and b = 2d.

(c2 + 2d)(c2 - 2d) = (c2)2 - (2d)2

= c4 - 4d2

Use the rule for (a + b)(a - b).

(a + b)(a - b) = a2 - b2

Identify a and b : a = 7 and b = y.

(7 + y)(7 - y) = 72 - y2

= 49 - y2 Understand the Problem.

The answer will be an expression that represents the area of the path.

List the important information :

(i) The pond is a square with a side length of (x - 3).

(ii) The path has a side length of (x + 3).

Make a Plan.

The area of the pond is (x - 3)2. The total area of the path plus the pond is (x + 3)2. You can subtract the area of the pond from the total area to find the area of the path.

Solve.

Step 1 :

Find the total area.

Use the rule for (a + b)2.

(a + b)2 = a2 + 2ab + b2

Identify a and b : a = x and b = 3.

(x + 3)2 = (x)2 + 2(x)(3) + (3)2

= x2 + 6x + 9

Step 2 :

Find the area of the pond.

Use the rule for (a - b)2.

(a - b)2 = a2 - 2ab + b2

Identify a and b : a = x and b = 3.

(x - 3)2 = (x)2 - 2(x)(3) + (3)2

= x2 - 6x + 9

Step 3 :

Find the area of the path.

Area of Path = Total Area - Area of Pond

= (x2 + 6x + 9) - (x2 - 6x + 9)

Use the Distributive Property.

= x2 + 6x + 9 - x2 + 6x - 9

Group like terms together.

= (x2 - x2) + (6x + 6x) + (9 - 9)

Combine like terms.

= 12x

The area of the path is 12x.

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