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Mathematics

Grade10

Easy

Question

Determine the independent term of x⁷ in the expansion of (3x² + 4)¹².

220 × 4⁶
230
548× 3!
 220 × 3⁶ × 4⁶

The correct answer is: 220 × 3⁶ × 4⁶

By using Binomial theorem
$sum for k equals 0 of open parentheses n to the power of k close parentheses x to the power of k y to the power of n times k end exponent equals n to the power of 0 x to the power of 0 y to the power of n plus n to the power of 1 x to the power of 1 y to the power of n times 1 end exponent plus n squared x squared y to the power of n times 2 end exponent plus horizontal ellipsis plus n to the power of n x to the power of n y to the power of 0 comma text where end text open parentheses n to the power of k close parentheses$
$equals fraction numerator n factorial over denominator k factorial left parenthesis n minus k right parenthesis factorial end fraction times text Now, end text T subscript r plus 1 end subscript equals C presuperscript n subscript r a to the power of n asterisk times r end exponent b to the power of r comma T subscript 9 plus 1 end subscript equals C presuperscript 12 subscript 6 a to the power of 12 minus 6 end exponent b to the power of 6 equals 220 asterisk times open parentheses 3 x squared close parentheses to the power of 6 asterisk times left parenthesis 4 right parenthesis to the power of 6 equals 220 asterisk times 3 to the power of 6 asterisk times 4 to the power of 6 times$
Hence the coefficient is $220 asterisk times 3 to the power of 6 asterisk times 4 to the power of 6$

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