Question
Use binomial theorem to expand (2c + d)6
Hint:
The binomial expansion is
, here n ≥ 0 .
We are asked to use binomial theorem to expand (2c + d)6 .
The correct answer is: values of the triangle
Step 1 of 2:
The given expression is (2c + d)6 , here x = 2c & y = d . The value of n=6, hence we would have 6+1=7 terms in the expression.
Step 2 of 2:
Substitute the values of (2c + d)6 in the binomial expression
![table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell left parenthesis 2 c plus d right parenthesis to the power of 6 equals 6 C subscript 0 left parenthesis 2 c right parenthesis to the power of 6 plus 6 C subscript 1 left parenthesis 2 c right parenthesis to the power of 5 left parenthesis d right parenthesis plus 6 C subscript 2 left parenthesis 2 c right parenthesis to the power of 4 left parenthesis d right parenthesis squared plus 6 C subscript 3 left parenthesis 2 c right parenthesis cubed left parenthesis d right parenthesis cubed plus 6 C subscript 4 left parenthesis 2 c right parenthesis squared left parenthesis d right parenthesis to the power of 4 plus 6 C subscript 5 left parenthesis 2 c right parenthesis left parenthesis d right parenthesis to the power of 5 plus 6 C subscript 6 left parenthesis d right parenthesis to the power of 6 end cell row cell equals 64 c to the power of 6 plus 6 open parentheses 32 c to the power of 5 close parentheses left parenthesis d right parenthesis plus 15 open parentheses 16 c to the power of 4 close parentheses open parentheses d squared close parentheses plus 20 open parentheses 8 c cubed close parentheses open parentheses d cubed close parentheses plus 15 open parentheses 4 c squared close parentheses open parentheses d to the power of 4 close parentheses plus 6 left parenthesis 2 c right parenthesis open parentheses d to the power of 5 close parentheses plus d to the power of 6 end cell row cell equals 64 c to the power of 6 plus 192 d c to the power of 5 plus 240 d squared c to the power of 4 plus 160 d cubed c cubed plus 60 d to the power of 4 c squared plus 12 d to the power of 5 c plus d to the power of 6 end cell end 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)
Thus, the expansion is:
.
Thus, the expansion is:
The expansion of the expression can also be found using the Pascal’s triangle. But, it is necessary to remember the values of the triangle to write down the expansion.
Related Questions to study
Use binomial theorem to expand
.
Use binomial theorem to expand
.
Use binomial theorem to expand (x - 1)7
For the expansion of an expression (x + y)n , we would have n+1 terms. This is something you need to keep in mind.
Use binomial theorem to expand (x - 1)7
For the expansion of an expression (x + y)n , we would have n+1 terms. This is something you need to keep in mind.
Use polynomial identities to factor the polynomials or simplify the expressions :
![x cubed minus 27 y cubed](data:image/png;base64,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)
Use polynomial identities to factor the polynomials or simplify the expressions :
![x cubed minus 27 y cubed](data:image/png;base64,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)
Use Pascal triangle to expand![left parenthesis x plus y right parenthesis to the power of 6](data:image/png;base64,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)
Use Pascal triangle to expand![left parenthesis x plus y right parenthesis to the power of 6](data:image/png;base64,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)
Use binomial theorem to expand (s2 + 3)5
For the expansion of an expression (x + y)n , we would have n+1 terms. This is something you need to keep in mind.
Use binomial theorem to expand (s2 + 3)5
For the expansion of an expression (x + y)n , we would have n+1 terms. This is something you need to keep in mind.
Use polynomial identities to factor the polynomials or simplify the expressions :
![8 x cubed plus y to the power of 9](data:image/png;base64,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)
Use polynomial identities to factor the polynomials or simplify the expressions :
![8 x cubed plus y to the power of 9](data:image/png;base64,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)
Use Pascal triangle to expand ![left parenthesis x plus y right parenthesis to the power of 6](data:image/png;base64,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)
Use Pascal triangle to expand ![left parenthesis x plus y right parenthesis to the power of 6](data:image/png;base64,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)
Use polynomial identities to factor the polynomials or simplify the expressions :
![x to the power of 9 minus 8](data:image/png;base64,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)
Use polynomial identities to factor the polynomials or simplify the expressions :
![x to the power of 9 minus 8](data:image/png;base64,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)
Use Pascal triangle to expand![left parenthesis x plus y right parenthesis to the power of 5](data:image/png;base64,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)
Use Pascal triangle to expand![left parenthesis x plus y right parenthesis to the power of 5](data:image/png;base64,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)
Use binomial theorem to expand (x - 3)4
For the expansion of an expression (x + y)n , we would have n+1 terms. This is something you need to keep in mind.
Use binomial theorem to expand (x - 3)4
For the expansion of an expression (x + y)n , we would have n+1 terms. This is something you need to keep in mind.
Use polynomial identities to factor the polynomials or simplify the expressions :
![x to the power of 8 minus 9](data:image/png;base64,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)
Use polynomial identities to factor the polynomials or simplify the expressions :
![x to the power of 8 minus 9](data:image/png;base64,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)
How is
obtained from ![left parenthesis x plus y right parenthesis to the power of n minus 1 end exponent](data:image/png;base64,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)
How is
obtained from ![left parenthesis x plus y right parenthesis to the power of n minus 1 end exponent](data:image/png;base64,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)
Use Pascal triangle to expand (x + y)6
The expansion of the polynomial (x + y)6 can be found using the values of 6Cr
Use Pascal triangle to expand (x + y)6
The expansion of the polynomial (x + y)6 can be found using the values of 6Cr