Problem 44 - Algebra 2

Question

In the expansion of \((m+n)^{9}\) , one of the terms contains \(m^{3}\) . a. What is the exponent of \(n\) in this term? b. What is the coefficient of this term?

Step-by-Step Solution

Answer

a. The exponent of \( n \) in the term \( m^{3} \) is 6. \n b. The coefficient of this term is 84.

Step 1: Find the term with \( m^{3} \)

The expression we have is \( (m + n)^{9} \). From the Binomial Theorem, the (r+1)th term in the expansion is \(\binom{9}{r} \cdot m^{9-r} \cdot n^{r}\). So, find out the value of 'r' for which the power of 'm' is 3. From \( m^{9-r} = m^{3} \) we can calculate that r=6.

Step 2: Find the exponent of \( n \)

The power of \( n \) in the term with \( m^{3} \) can now be easily computed using value of r. It is equal to \( n^{r} = n^{6} \). So, the exponent of \( n \) in the term \( m^{3} \) is 6.

Step 3: Calculate the Coefficient of the Term.

The coefficient of the term is given by \(\binom{9}{r}\), which in our case is \(\binom{9}{6}\). This equals to 84. Hence, the coefficient of the term is 84.

Key Concepts

Understanding Exponents

Exponents, also known as powers, represent how many times a number, called the base, is multiplied by itself. In our original problem, when we see \( (m+n)^9 \), the number 9 is the exponent. This tells us that \( (m+n) \) is multiplied by itself 9 times.

When dealing with exponents, there are a few important rules to remember:

• Multiplication: \( a^m imes a^n = a^{m+n} \)
• Division: \( \frac{a^m}{a^n} = a^{m-n} \) for \( a eq 0 \)
• Zero Exponent: \( a^0 = 1 \) for \( a eq 0 \)
• Power of a Power: \( (a^m)^n = a^{m \times n} \)

In the term from the binomial expansion of \( (m+n)^9 \) where \( m^3 \) appears, we used the rule that tells us to find the specific exponent for each part of the expanded term.

Coefficients Explained

Coefficients are the numbers that multiply the variables in algebra. In a term like \( 5x^2 \), the coefficient is 5. They are very important in algebraic expressions and expansions, as they provide the multiplier for each term.

When expanding using the Binomial Theorem, the coefficients can be found using binomial coefficients, which are calculated using the formula:

• \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

This represents the number of ways to choose 'r' elements from a set of 'n' elements without considering the order. These coefficients essentially tell us how strong each term is in the expression. In our exercise, the coefficient found for the term \( m^3 \) was determined to be 84, using the binomial coefficient \( \binom{9}{6} = 84 \). This shows the balancing role of coefficients in expressing the weighted sum of terms in polynomial expansions.

Explaining Polynomial Expansion

Polynomial expansion is breaking down expressions like \( (m+n)^9 \) into a sum of terms. This process uses the Binomial Theorem to expand expressions of the form \( (a + b)^n \).

The Binomial Theorem states:

• \[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]

This means each term in the expansion is composed of a binomial coefficient multiplied by the appropriate powers of the two terms. For \( (m+n)^9 \), this expansion lets us find specific terms such as the one including \( m^3 \). Knowing how to expand polynomials allows you to handle complex algebraic expressions, simplify calculations, and solve for variables in an easy, organized manner.

By mastering polynomial expansion concepts, you gain the ability to simplify complex problems and express complicated expressions more manageably. This exercise of calculating specific terms like \( m^3 n^6 \) demonstrates the real-world utility of understanding polynomial expansion.