Mathematics
If a + b = 8 and ab = 15, find the value of a4 + a2b2 + b4
Factorisation
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Answer
Given,
a + b = 8 and ab = 15
∴ (a + b)2 = 82
⇒ a2 + b2 + 2ab = 64
⇒ a2 + b2 + 2(15) = 64
⇒ a2 + b2 = 64 - 30 = 34.
a4 + a2b2 + b4 = a4 + 2a2b2 - a2b2 + b4
= (a2 + b2)2 - a2b2
We know that,
a2 - b2 = (a + b)(a - b)
∴ (a2 + b2)2 - a2b2 = (a2 + b2 + ab)(a2 + b2 - ab)
= (34 + 15)(34 - 15)
= 49 × 19
= 931.
Hence, the value of a4 + a2b2 + b4 = 931.
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