How to Deduce the Coefficient of x^n in the Expansion of (x+3)(1+2x)^(-2)?

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In summary, the conversation discusses finding the coefficients of the x^n term in the expansion of (1+2x)^(-2) and (x+3)(1+2x)^(-2). The first 4 terms of the first expansion are given as 1 - 4x + 12x^2 - 32x^3 + ... and the coefficient of the x^n term in the second expansion is [(-1)^n](n+1)(2^n). The last part involves deducing the coefficient of the x^n term, which can be found by multiplying the x^n term by an x.
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Homework Statement


By finding the first 4 terms in the expansion of (1+2x)^(-2) in ascending powers of x and find the coefficient of x^n term. Hence deduce the coefficient of x^n term in the expansion of (x+3)(1+2x)^(-2).


Homework Equations





The Attempt at a Solution


The first two parts were easy. For the first part, the answer is 1 - 4x + 12x^2 - 32x^3 + ...
For the second part, [(-1)^n](n+1)(2^n). But I'm stunned at the last part. How do I deduce?
 
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Well the only term that will give you an 'x' term is if you multiply an x by an x. So if you multiply the xn term by the x, you will get the nth term for the expansion.
 

FAQ: How to Deduce the Coefficient of x^n in the Expansion of (x+3)(1+2x)^(-2)?

What does "deduce the coefficient of x^n" mean?

"Deducing the coefficient of x^n" refers to finding the numerical value of the coefficient that multiplies the variable x raised to the power of n in a given polynomial expression.

Why is it important to deduce the coefficient of x^n?

Knowing the coefficient of x^n is important in determining the behavior and characteristics of a polynomial function. It can help in graphing the function, finding its roots, and solving equations involving the function.

How do I deduce the coefficient of x^n?

To deduce the coefficient of x^n, you can use the method of equating coefficients or the method of substitution. In both methods, you will need to have a polynomial expression with the variable x raised to the power of n, and a known value for the expression.

What is the difference between deducing the coefficient of x^n and finding the value of x^n?

Deducing the coefficient of x^n involves finding the numerical value of the coefficient itself, while finding the value of x^n involves evaluating the expression with a specific value for x. Deducing the coefficient is more focused on the structure of the polynomial expression, while finding the value is more focused on the output of the expression.

Can the coefficient of x^n be a negative number?

Yes, the coefficient of x^n can be a negative number. It depends on the given polynomial expression and the value of n. The coefficient can be positive, negative, or zero. It is important to pay attention to the signs and values of coefficients when deducing them.

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