In this section, you will learn the formula or expansion for (a + b)^{3}.

That is,

(a + b)^{3} = (a + b)(a + b)(a + b)

Multiply (a + b) and (a + b).

(a + b)^{3} = (a^{2} + ab + ab + b^{2})(a + b)

Simplify.

(a + b)^{3} = (a^{2} + 2ab + b^{2})(a + b)

(a + b)^{3} = a^{3} + a^{2}b + 2a^{2}b + 2ab^{2} + ab^{2} + b^{3}

Combine the like terms.

**(a + b) ^{3} = a^{3} + 3a^{2}b + 3ab^{2} + b^{3}**

or

**(a + b) ^{3} = a^{3} + b^{3} + 3ab(a + b) **

**Problem 1 :**

Expand :

(x + 2)^{3 }

**Solution :**

(x + 2)^{3 }is in the form of (a + b)^{3}

Comparing (a + b)^{3 }and (x + 2)^{3}, we get

a = x

b = 2

Write the formula / expansion for (a + b)^{3}.

(a + b)^{3} = a^{3} + 3a^{2}b + 3ab^{2} + b^{3}

Substitute x for a and 2 for b.

(x + 2)^{3} = x^{3} + 3(x^{2})(2) + 3(x)(2^{2}) + 2^{3}

(x + 2)^{3} = x^{3} + 6x^{2} + 3(x)(4) + 8

(x + 2)^{3} = x^{3} + 6x^{2} + 12x + 8

So, the expansion of (x + 2)^{3} is

x^{3} + 6x^{2} + 12x + 8

**Problem 2 :**

Expand :

(2x + 3)^{3 }

**Solution :**

(2x + 3)^{3 }is in the form of (a + b)^{3}

Comparing (a + b)^{3 }and (2x + 3)^{3}, we get

a = 2x

b = 3

Write the formula / expansion for (a + b)^{3}.

(a + b)^{3} = a^{3} + 3a^{2}b + 3ab^{2} + b^{3}

Substitute 2x for a and 3 for b.

(2x + 3)^{3} = (2x)^{3} + 3(2x)^{2}(3) + 3(2x)(3^{2}) + 3^{3}

(2x + 3)^{3} = 8x^{3} + 3(4x^{2})(3) + 3(2x)(9) + 27

(2x + 3)^{3} = 8x^{3} + 36x^{2} + 54x + 27

So, the expansion of (2x + 3)^{3} is

8x^{3} + 36x^{2} + 54x + 27

**Problem 3 :**

Expand :

(p + 2q)^{3 }

**Solution :**

(p + 2q)^{3 }is in the form of (a + b)^{3}

Comparing (a + b)^{3 }and (p + 2q)^{3}, we get

a = p

b = 2q

Write the formula / expansion for (a + b)^{3}.

(a + b)^{3} = a^{3} + 3a^{2}b + 3ab^{2} + b^{3}

Substitute p for a and 2q for b.

(p + 2q)^{3} = p^{3} + 3(p^{2})(2q) + 3(p)(2q)^{2} + (2q)^{3}

(p + 2q)^{3} = p^{3} + 6p^{2}q + 3(p)(4q^{2}) + 8q^{3}

(p + 2q)^{3} = p^{3} + 6p^{2}q + 12pq^{2} + 8q^{3}

So, the expansion of (p + 2q)^{3} is

p^{3} + 6p^{2}q + 12pq^{2} + 8q^{3}

**Problem 4 : **

If a + b = 12 and a^{3} + b^{3} = 468, then find the value of ab.^{ }

**Solution :**

To find the value of ab, we can use the formula or expansion for (a + b)^{3}.

Write the formula / expansion for (a + b)^{3}.

(a + b)^{3} = a^{3} + 3a^{2}b + 3ab^{2} + b^{3}

or

(a + b)^{3} = a^{3} + b^{3} + 3ab(a + b)

Substitute 12 for (a + b) and 468 for (a^{3} + b^{3}).

(12)^{3} = 468 + 3(ab)(12)

Simplify.

1728 = 468 + 36ab

Subtract 468 from each side.

1260 = 36ab

Divide each side by 36.

35 = ab

So, the value of ab is 35.

**Problem 5 :**

Find the value of :

(107)^{3}

**Solution :**

We can use the algebraic formula for (a + b)^{3 }and find the value of (107)^{3 }easily.

Write (107)^{3 }in the form of (a + b)^{3}.

(107)^{3} = (100 + 7)^{3}

Write the expansion for (a + b)^{3}.

(a + b)^{3} = a^{3} + b^{3} + 3ab(a + b)

Substitute 100 for a and 7 for b.

(100 + 7)^{3} = 100^{3} + 7^{3} + 3(100)(7)(100 + 7)

(100 + 7)^{3} = 1000000 + 343 + 3(100)(7)(107)

(100 + 7)^{3} = 1000000 + 343 + 224700

(107)^{3} = 1225043

So, the value of (107)^{3} is

1,225,043

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