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 = (a2 + ab + ab + b2)(a + b)
Simplify.
(a + b)3 = (a2 + 2ab + b2)(a + b)
(a + b)3 = a3 + a2b + 2a2b + 2ab2 + ab2 + b3
Combine the like terms.
(a + b)3 = a3 + 3a2b + 3ab2 + b3
or
(a + b)3 = a3 + b3 + 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 = a3 + 3a2b + 3ab2 + b3
Substitute x for a and 2 for b.
(x + 2)3 = x3 + 3(x2)(2) + 3(x)(22) + 23
(x + 2)3 = x3 + 6x2 + 3(x)(4) + 8
(x + 2)3 = x3 + 6x2 + 12x + 8
So, the expansion of (x + 2)3 is
x3 + 6x2 + 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 = a3 + 3a2b + 3ab2 + b3
Substitute 2x for a and 3 for b.
(2x + 3)3 = (2x)3 + 3(2x)2(3) + 3(2x)(32) + 33
(2x + 3)3 = 8x3 + 3(4x2)(3) + 3(2x)(9) + 27
(2x + 3)3 = 8x3 + 36x2 + 54x + 27
So, the expansion of (2x + 3)3 is
8x3 + 36x2 + 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 = a3 + 3a2b + 3ab2 + b3
Substitute p for a and 2q for b.
(p + 2q)3 = p3 + 3(p2)(2q) + 3(p)(2q)2 + (2q)3
(p + 2q)3 = p3 + 6p2q + 3(p)(4q2) + 8q3
(p + 2q)3 = p3 + 6p2q + 12pq2 + 8q3
So, the expansion of (p + 2q)3 is
p3 + 6p2q + 12pq2 + 8q3
Problem 4 :
If a + b = 12 and a3 + b3 = 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 = a3 + 3a2b + 3ab2 + b3
or
(a + b)3 = a3 + b3 + 3ab(a + b)
Substitute 12 for (a + b) and 468 for (a3 + b3).
(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 = a3 + b3 + 3ab(a + b)
Substitute 100 for a and 7 for b.
(100 + 7)3 = 1003 + 73 + 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
Apart from the stuff explained above, if you would like to learn about more identities in Algebra,
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