Problem 40 - Algebra and Trigonometry with Analytic Geometry

Question

Express as a polynomial. $$ (3 x-4 y)^{3} $$

Step-by-Step Solution

Answer

The expanded polynomial is \(27x^3 - 108x^2y + 144xy^2 - 64y^3\).

Step 1: Identify the Expression

The given exercise is to express \[(3x - 4y)^3\] as a polynomial. This means expanding the expression and simplifying it to a standard polynomial form.

Step 2: Apply the Binomial Theorem

The binomial theorem states \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]. Here, we will use it to expand \[(3x - 4y)^3\], where \(a = 3x\), \(b = -4y\), and \(n = 3\). This involves calculating each term of the expansion.

Step 3: Calculate Each Term of the Expansion

For each term \(k\) from 0 to 3, calculate:- \(k = 0\): \(\binom{3}{0}(3x)^{3-0}(-4y)^0 = 1 \, \cdot \, 27x^3 \, \cdot \, 1 = 27x^3\)- \(k = 1\): \(\binom{3}{1}(3x)^{3-1}(-4y)^1 = 3 \, \cdot \, 9x^{2} \, \cdot \, (-4y) = -108x^{2}y\)- \(k = 2\): \(\binom{3}{2}(3x)^{3-2}(-4y)^2 = 3 \, \cdot \, 3x \, \cdot \, 16y^{2} = 144xy^{2}\)- \(k = 3\): \(\binom{3}{3}(3x)^{3-3}(-4y)^3 = 1 \, \cdot \, 1 \, \cdot \, -64y^3 = -64y^3\)

Key Concepts

Binomial Theorem

The binomial theorem is an essential tool in algebra for expanding expressions that are raised to a power. It might seem a bit complex at first, but breaking it down step-by-step can make it easier to understand. Essentially, the theorem gives a formula that describes the expansion of \((a + b)^n\). Each term in the expansion involves coefficients known as binomial coefficients, which are found using combinations. These coefficients can be calculated using the formula \(\binom{n}{k}\), which can also be read as "n choose k."

• \((a+b)^n\) expands into a sum of terms.
• Each term is a product of a binomial coefficient, a power of \(a\), and a power of \(b\).
• The sum covers all terms from \(k=0\) to \(k=n\).

In the exercise, using the binomial theorem helps expand \((3x - 4y)^3\) into a polynomial by computing each term separately. This is done by identifying \(a = 3x\), \(b = -4y\), and \(n = 3\), then substituting these into the binomial theorem formula.

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating those symbols. It's like a universal language for math that helps express relationships and solve equations. By using algebra, we can structure complex expressions in simpler terms.
The problem at hand involves algebraic manipulation to expand the expression \((3x - 4y)^3\) into a polynomial form. This involves recognizing and applying algebraic identities, such as the binomial theorem.

• We substitute terms like \(3x\) and \(-4y\) into algebraic formulas.
• We compute powers and products to convert algebraic expressions into their expanded forms.
• Algebraic simplification helps in transforming expressions into more manageable forms.

In this context, algebra aids in organizing and applying the math rules that turn the original bracketed expression into a simpler polynomial.

Polynomial Simplification

Polynomial simplification is the process of making a polynomial expression as concise as possible. It involves combining like terms and ordering them, usually, by the degree of their terms.
Once each term of the binomial has been expanded using the binomial theorem, the final step is to simplify it into its polynomial form. Simplification includes:

• Identifying like terms: For polynomials, terms are alike if they have the same variable parts with the same exponents.
• Combining like terms: Add or subtract coefficients of like terms to simplify the polynomial.
• Sorting by degree: The polynomial is often reordered starting with the highest exponent.

With the example \((3x - 4y)^3\), after expanding using the binomial theorem, simplify by writing it as \(27x^3 - 108x^2y + 144xy^2 - 64y^3\). This expression is neatly simplified and is the polynomial form of the original expression.