Question

1. expand each binomial on the left by applying pascal’s triangle and then match it to its correct expansion on the right.
2. ((x + 2y)^3)
3. ((2 + y)^3)
4. ((-2 + y)^3)
5. ((x - 2y)^3)

a. (-8 + 12y - 6y^2 + y^3)
b. (x^3 - 6x^2y + 12xy^2 - 8y^3)
c. (x^3 + 6x^2y + 12xy^2 + 8y^3)
d. (8 + 12y + 6y^2 + y^3)

Explanation:

To solve this, we use the binomial expansion formula \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\) (or \((a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\) for subtraction).

Step 1: Expand \((x + 2y)^3\)

Let \(a = x\), \(b = 2y\).

$$
LATEXBLOCK0
$$

Step 2: Expand \((2 + y)^3\)

Let \(a = 2\), \(b = y\).

$$
LATEXBLOCK1
$$

Step 3: Expand \((-2 + y)^3\)

Let \(a = -2\), \(b = y\).

$$
LATEXBLOCK2
$$

Step 4: Expand \((x - 2y)^3\)

Let \(a = x\), \(b = 2y\) (using \((a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\)).

$$
LATEXBLOCK3
$$

Answer:

1. \((x + 2y)^3\) - c. \(x^3 + 6x^2y + 12xy^2 + 8y^3\)
2. \((2 + y)^3\) - d. \(8 + 12y + 6y^2 + y^3\)
3. \((-2 + y)^3\) - a. \(-8 + 12y - 6y^2 + y^3\)
4. \((x - 2y)^3\) - b. \(x^3 - 6x^2y + 12xy^2 - 8y^3\)