Find the coefficient of x⁴ in (x²+x+1)+(x²-x+1)
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♣ Qᴜᴇꜱᴛɪᴏɴ :
Find the coefficient of x⁴ in (x²+x+1)²+(x²-x+1)²
♣ ɢɪᴠᴇɴ :
(x²+x+1)²+(x²-x+1)²
♣ ᴛᴏ ꜰɪɴᴅ :
Coefficient of x⁴
♣ ᴀɴꜱᴡᴇʀ :
In ( x²+x+1)²+(x²-x+1)², coeffiecient of x² = 2
♣ ᴄᴀʟᴄᴜʟᴀᴛɪᴏɴꜱ :
(x²+x+1)²+(x²-x+1)²
= (x²+x+1)²+(x²-x+1)²
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Using algebraic identity : (a+b+c)² = a² + b² + c² + 2(ab + bc + ca)
(x²+x+1)² = x⁴ + x² + 1 + 2(x²x + 1x +1x²)
Using algebraic identity : (a-b+c)² =a² + b² + c² - 2(ab + bc -ac)
(x²-x+1)² = x⁴ + x² + 1 - 2(x²x + 1x - 1x²)
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(x²+x+1)²+(x²-x+1)²
= x⁴ + x² + 1 + 2(x²x + 1x + 1x²) + x⁴ + x² + 1 - 2(x²x + 1x - 1x²)
= x⁴ + x² + 1 + 2x²x + 2x + 2x² + x⁴ + x² + 1 - 2x²x + 2x - 2x²
= 2x⁴ + 2x² +2
A coefficient is a number multiplied by a variable.
In ( x²+x+1)²+(x²-x+1)², coeffiecient of x⁴ :
= 2
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