Problem 70 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the coefficient \(a\) of the given term in the expansion of the binomial. Binomial = \(\left(z^{2}+y\right)^{12}\) Term = \(a z^{14} y^{5}\)

Step-by-Step Solution

Answer

The coefficient \(a\) of the term \(a z^{14} y^{5}\) in the binomial expansion is 792.

Step 1 Identify the binomial, the power and the term

In this case, the binomial to expand is \(\left(z^{2}+y\right)\), where \(x=z^2\) and \(y=y\). The power of the binomial (n) is 12. The term to find is \(a z^{14} y^{5}\). The coefficients can be obtained from the formula \(_nC_r x^{n-r} y^r\). We need to find out the value of \(a\).

Step 2 Use Binomial Theorem

The coefficient \(a\) can be obtained from the formula \(_nC_r x^{n-r} y^r\), where \(n=12\), \(r\) is the power of \(y\) in the term to match \(y^5\), \(x=z^2\) and \(y=y\). We need to find a value of \(r\) such that \(n-r\) is equal to the power of \(z^2\) in \(z^{14}\). After this, multiply the binomial coefficient by \(x^{n-r} y^r\).

Step 3 Find the binomial coefficient and use it to find the coefficient

Having \(n-r = \frac{14}{2} = 7\), we can find \(r=n-7=12-7=5\). So, \(r=5\). The binomial coefficient \(_nC_r\) is equal to \(_{12}C_{5}\) or '12 choose 5'. The value is obtained by \(\frac{12!}{5!(12-5)!}\) which equals 792. Hence, the coefficient \(a\) for the term \(a z^{14} y^{5}\) is 792.

Key Concepts

Binomial coefficient

The binomial coefficient is a key component in the Binomial Theorem, which is used to expand expressions that are raised to a power. In our problem, the binomial coefficient is written as \(_{n}C_{r}\), where \(n\) is the total number of items, and \(r\) is the number of items to choose. It is sometimes spoken as "n choose r." This coefficient tells us how many ways we can choose \(r\) items out of \(n\), and is calculated using the formula:

• \(\frac{n!}{r!(n-r)!}\)

"!" is called factorial and it means to multiply a series of descending natural numbers. For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).
To find the coefficient in the expansion problem, we used \(n=12\) and \(r=5\), because the term we were looking for requires the power of \(y\) to be 5. The resulting coefficient \(_{12}C_{5}\) equals 792, indicating there are 792 ways to pick 5 elements out of 12.

Polynomial expansion

Polynomial expansion is a technique used to express a polynomial raised to a certain power as a sum of terms. These terms are formed by applying the Binomial Theorem, which states:

• \((x+y)^n = \sum_{r=0}^{n} \binom{n}{r} x^{n-r} y^r\)

This concept helps us take something like \((z^2 + y)^{12}\) and break it down into simpler monomials such as \(z^{14}y^5\), with various coefficients in front.
Each term in the expanded form is derived using different values of \(r\) until all powers add up correctly to the overall degree of the polymonial, originally given by \(n\). The coefficient of each term is dictated by the binomial coefficient, which ensures each piece of the expanded polynomial matches the original exponent values.

Algebra

Algebra is the branch of mathematics that deals with symbols and the rules for manipulating those symbols. It is a unifying thread of almost all mathematics and plays a crucial role when dealing with polynomial expansions.
In this problem, algebraic rules help in understanding how to select which 'powers' or exponents we need for terms\(z^{14}y^5\) from \((z^2+y)^{12}\). It allows us to solve complex problems by breaking them down into understandable parts, using methods like substituting powers \(z^2\) and arranging terms using the binomial theorem.
Understanding algebra ensures that you can manipulate expressions, such as factoring or expanding, and helps in performing operations to find desired results like coefficients, such as in this binomial expansion.