Problem 40
Question
If the coefficient of \(x^{7}\) in \(\left[a x^{2}+\left(\frac{1}{b x}\right)\right]^{11}\) equals the coefficient of \(x^{-7}\) in \(\left[a x-\left(\frac{1}{b x^{2}}\right)\right]^{11}\), then a and bsatisfy the relation (a) \(a-b=1\) (b) \(a+\underline{b}=1\) (c) \(\frac{a}{b}=1\) (d) \(a b=1\)
Step-by-Step Solution
Verified Answer
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1Step 1: Identify Binomial Coefficients
To find the coefficient of a particular term in a binomial expression of the form \((ax^m + bx^n)^p\), use the general term formula \(T_{r+1} = \binom{p}{r} (ax^m)^{p-r} (bx^n)^{r}\). We need to apply this formula to each binomial expression given in the problem.
2Step 2: Calculate Coefficient of \(x^7\) in First Expression
For the first expression \(\left[a x^{2} + \frac{1}{b x}\right]^{11}\), we use the general term formula. The term is \(\binom{11}{r} (a x^2)^{11-r} \left(\frac{1}{b x}\right)^{r}\), leading to \(x^{2(11-r)-r}\). Set this equal to 7: \[2(11-r) - r = 7\]Solving, we get: \[22 - 3r = 7 \implies 3r = 15 \implies r = 5\].Substitute \(r = 5\) back to find the coefficient: \[\binom{11}{5} a^{6} \left(\frac{1}{b}\right)^{5}\].
3Step 3: Calculate Coefficient of \(x^{-7}\) in Second Expression
For the second expression \(\left[a x - \frac{1}{b x^2}\right]^{11}\), the general term is \(\binom{11}{r} (a x)^{11-r} \left(-\frac{1}{b x^2}\right)^{r}\), leading to \(x^{11-r-2r}\). Set this equal to -7: \[11 - 3r = -7\]Solving gives: \[11 - 3r = -7 \implies 3r = 18 \implies r = 6\].Substitute \(r = 6\) to find the coefficient: \[\binom{11}{6} a^{5} \left(-\frac{1}{b}\right)^{6}\].
4Step 4: Equate Coefficients and Solve for Relation
Equate the coefficients found in Steps 2 and 3: \[\binom{11}{5} a^6 \left(\frac{1}{b}\right)^5 = \binom{11}{6} a^5 \left(-\frac{1}{b}\right)^6\].Simplify: \[a^6 \cdot b^6 = a^5 \cdot b^5 \cdot (-b)\]This yields the equation \[a \cdot -b = 1\] or \(a b = -1\). Since multiplication of integers does not typically yield negative if both are positive, inspect carefully or notice dimensional coefficients sign direction cancellation in exam specifics wherein match to option via...
Key Concepts
Binomial CoefficientsExponentiationPolynomial Equations
Binomial Coefficients
Binomial coefficients are at the heart of the binomial theorem. They are the numerical factors that multiply the terms in a binomial expansion. For a binomial raised to the power of "n", these coefficients are represented as \( \binom{n}{k} \), which read as "n choose k". This representation calculates the number of ways to choose "k" elements from a set of "n" elements without regard to order. In the binomial formula, these coefficients determine the weight of each term in the expansion.
To intuitively understand binomial coefficients, consider a simple binomial \((a + b)^n\). In its expansion, each term will have a binomial coefficient, which can be calculated using the formula for combinations:
\[\binom{n}{k} = \frac{n!}{k!(n-k)!} \]
where "!" denotes factorial, meaning that you multiply the number by every positive integer less than itself. Applying these principles helps in determining specific term coefficients much like in the example we analyzed where different terms of a polynomial expression required coefficient calculations.
To intuitively understand binomial coefficients, consider a simple binomial \((a + b)^n\). In its expansion, each term will have a binomial coefficient, which can be calculated using the formula for combinations:
\[\binom{n}{k} = \frac{n!}{k!(n-k)!} \]
where "!" denotes factorial, meaning that you multiply the number by every positive integer less than itself. Applying these principles helps in determining specific term coefficients much like in the example we analyzed where different terms of a polynomial expression required coefficient calculations.
Exponentiation
Exponentiation is a fundamental operation in algebra, representing repeated multiplication of a base number. When you see an expression like \(b^n\), it means you multiply "b" by itself "n" times. In the context of polynomials, exponentiation defines the degree of a term, demonstrated by the addition of exponents in expressions with the same bases.
In our specific problem involving binomials, exponentiation plays a crucial role. The power to which a binomial is raised determines the number of terms in the resulting polynomial when expanded. Here, understanding of very large exponents in binomial expansions, ranging up to the 11th power, was required to tackle the given problem. This capability allows for the manipulation of terms and coefficients to achieve desired expressions, in this case determining if two specific polynomial coefficients are equal.
In our specific problem involving binomials, exponentiation plays a crucial role. The power to which a binomial is raised determines the number of terms in the resulting polynomial when expanded. Here, understanding of very large exponents in binomial expansions, ranging up to the 11th power, was required to tackle the given problem. This capability allows for the manipulation of terms and coefficients to achieve desired expressions, in this case determining if two specific polynomial coefficients are equal.
Polynomial Equations
Polynomial equations are mathematical expressions involving sums of powers in one or more variables multiplied by coefficients. A polynomial is characterized by its degree—which is determined by the highest power of the variable within the expression.
When solving polynomial equations, one often rearranges terms and combines like terms to reduce the equation to a simpler form. Here, the complexity might involve multiple steps as seen in binomial expansions where terms of varying exponents and coefficients are combined and simplified.
The exercise involving binomials translated into solving polynomial equations showed how terms with specific coefficients matched at particular powers—like \(x^7\) and \(x^{-7}\) in this case. These require specific techniques and keen observation to solve by equating terms and resolving through algebraic manipulation to understand the relationship between variables—often revealing solutions to the components of the expressions such as the satisfying relation \(a \cdot b = -1\) discussed in the problem.
When solving polynomial equations, one often rearranges terms and combines like terms to reduce the equation to a simpler form. Here, the complexity might involve multiple steps as seen in binomial expansions where terms of varying exponents and coefficients are combined and simplified.
The exercise involving binomials translated into solving polynomial equations showed how terms with specific coefficients matched at particular powers—like \(x^7\) and \(x^{-7}\) in this case. These require specific techniques and keen observation to solve by equating terms and resolving through algebraic manipulation to understand the relationship between variables—often revealing solutions to the components of the expressions such as the satisfying relation \(a \cdot b = -1\) discussed in the problem.
Other exercises in this chapter
Problem 38
In the binomial expansion of \((a-b)^{n}, n \geq 5\), the sum of \(5^{\text {th }}\) and \(6^{\text {th }}\) terms is zero, then \(\mathrm{a} / \mathrm{b}\) equ
View solution Problem 39
For natural numbers \(\mathrm{m}, \mathrm{n}\) if \((1-y)^{m}(1+y)^{n}\) \(=1+a_{1} y+a_{2} y^{2}+\ldots . . .\) and \(a_{1}=a_{2}=10\), then \((m, n)\) is \(\b
View solution Problem 41
The coefficient of \(x^{n}\) in expansion of \((1+x)(1-x)^{n}\) is (a) \((-1)^{n-1} n\) (b) \((-1)^{n}(1-n)\) (c) \((-1)^{n-1}(n-1)^{2}\) (d) \((n-1)\)
View solution Problem 42
The number of integral terms in the expansion of \((\sqrt{3}+\sqrt[8]{5})^{256}\) is (a) 35 (b) 32 (c) 33 (d) 34
View solution