Problem 44 - Algebra and Trigonometry

Question

Perform the indicated operations and simplify. $$ (1-2 y)^{3} $$

Step-by-Step Solution

Answer

\(-8y^3 + 12y^2 - 6y + 1\)

Step 1: Understanding the Expression

We need to expand and simplify the expression \((1 - 2y)^3\). This is a binomial expression raised to the power of 3.

Step 2: Apply the Binomial Theorem

According to the Binomial Theorem, \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Here, \(a = 1\), \(b = -2y\), and \(n = 3\).

Step 3: Calculate Each Term

The expansion according to the Binomial Theorem will have 4 terms: 1. \(\binom{3}{0} (1)^{3-0} (-2y)^0 = 1\)2. \(\binom{3}{1} (1)^{3-1} (-2y)^1 = 3 \times (-2y) = -6y\)3. \(\binom{3}{2} (1)^{3-2} (-2y)^2 = 3 \times (4y^2) = 12y^2\)4. \(\binom{3}{3} (1)^{3-3} (-2y)^3 = (-8y^3) = -8y^3\)

Step 4: Combine the Terms

Now we add all the terms we found: \(1 - 6y + 12y^2 - 8y^3\).

Step 5: Simplify the Expression

The simplified expression is: \(-8y^3 + 12y^2 - 6y + 1\).

Key Concepts

Algebraic Expressions

Algebraic expressions are like mathematical phrases that use numbers, variables, and operations. These serve as "building blocks" to create meaningful calculations and equations. An expression can be as simple as a number, like 3, or as complicated as \(1 - 2y\)^3, a binomial expression like the one in the exercise.

• **Components:** Each expression might include numbers (constants) and variables (unknown values represented by letters).
• **Operations:** The operations used in expressions include addition, subtraction, multiplication, division, and exponents.
• **Terms:** A term is a single mathematical expression component, separated by a plus or minus sign. In \(1 - 2y\)^3, "1" and "-2y" are terms.

Algebraic expressions are foundational in algebra. They allow us to manipulate and solve problems using various techniques, including simplifying, factoring, and expanding, like in our problem with \(1 - 2y\)^3.

Polynomials

Polynomials are algebraic expressions that consist of multiple terms combined using addition and subtraction. They are a more specific type of algebraic expression. A polynomial like the result from our exercise \(-8y^3 + 12y^2 - 6y + 1\) is composed of several terms based on powers of a variable, such as "y" in this case.

• **Degree:** The degree of a polynomial is the highest exponent of the variable. For example, \(-8y^3\) has a degree of 3.
• **Terms:** Each polynomial is made up of terms. The number of terms varies but includes constants and variables raised to a power, like \12y^2\.
• **Standard Form:** Polynomials are usually written in a standard form where terms are ordered from highest to lowest degree. Hence, \(-8y^3 + 12y^2 - 6y + 1\) is in standard form.

Understanding polynomials goes beyond simple operations. It involves recognizing patterns and structures to simplify equations or expressions.

Binomial Expansion

Binomial expansion is a technique to expand expressions raised to a power, shown by the exercise \(1 - 2y\)^3. The binomial theorem is crucial here, offering a formula to expand.

• **Theorem:** The binomial theorem formula \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\) helps in systematically expanding binomials.
• **Coefficients:** Calculating these involves using combinations, represented by \binom{n}{k}\. The results give coefficients of terms in the expanded form.
• **Terms by Terms Calculation:** Applying the formula calculates each term in the sequence, like finding \(\binom{3}{2} (1)^{1} (-2y)^2 = 12y^2\).

Binomial expansion saves effort and ensures accuracy in expanding binomials. It's not just about multiplying terms, but also about understanding how each term builds on the previous one to complete the expansion.