Problem 43 - Algebra and Trigonometry with Analytic Geometry

Question

Express as a polynomial. $$ (2 x+y-3 z)^{2} $$

Step-by-Step Solution

Answer

The polynomial is \(4x^2 + y^2 + 9z^2 + 4xy - 6yz - 12xz\).

Step 1: Understand the Expression

The given expression is \((2x+y-3z)^2\). It is a binomial expression raised to the power of 2, which means we need to square the terms inside the parentheses.

Step 2: Apply the Binomial Theorem

The binomial theorem states that \((a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca\). Here, identify \(a = 2x\), \(b = y\), and \(c = -3z\).

Step 3: Calculate Individual Squares

Calculate each squared term: \((2x)^2 = 4x^2\), \(y^2 = y^2\), and \((-3z)^2 = 9z^2\).

Step 4: Calculate the Cross Terms

Calculate the products of the cross terms: \(2(2x)(y) = 4xy\), \(2(y)(-3z) = -6yz\), and \(2(2x)(-3z) = -12xz\).

Step 5: Combine All Terms

Combine all the terms to form the polynomial: \(4x^2 + y^2 + 9z^2 + 4xy - 6yz - 12xz\).

Key Concepts

Binomial Theorem

The binomial theorem is a powerful tool in algebra that helps us expand expressions that are raised to a given power. In its simplest form, it deals with expressions like

• \((a + b)^n\),
• where \(a\) and \(b\) are any expressions, and \(n\) is a non-negative integer.

Using the binomial theorem, we can expand these expressions without lengthy multiplication. When there are three terms involved, like in our problem \((2x+y-3z)^2\), we apply a variation of the theorem. Simply put, the theorem provides a way to systematically expand \((a + b + c)^n\).This involves finding the square of each term and combining them with twice the product of each possible pair.

• For our problem, the components are
  • \(a = 2x\),
  • \(b = y\), and
  • \(c = -3z\).

When expanded, it gives a polynomial consisting of several terms formed by squaring and multiplying these components.

Squaring Binomials

Squaring binomial expressions is a fundamental skill in algebra that allows us to understand polynomial relationships better. When we square a binomial,

• we essentially multiply the binomial by itself,

which means \((a + b)^2 = (a + b)(a + b) = a^2 + 2ab + b^2\). Think of it like creating a grid: each term in the binomial multiplies with every other term. In our initial problem of \((2x+y-3z)^2\), each term interacts with every other term and itself. The result includes:

• the square of each individual term, and
• twice the product of each possible pair of terms.

This method of expanding three-term expressions, while slightly more complex than two-terms, follows a similar principle

• which involves systematically applying the expansion method.

For the expression given, squaring results in terms like

• \((2x)^2\),
• \((y)^2\),
• \((-3z)^2\),
• as well as interaction terms
  • like \(2(2x)(y)\),
  • \(2(y)(-3z)\), and
  • \(2(2x)(-3z)\).

Algebraic Expressions

Algebraic expressions are the building blocks of algebra and involve combinations of numbers, variables, and operations. They allow us to describe general relationships and solve various mathematical problems.An algebraic expression can encompass terms connected through operators such as addition, subtraction, multiplication, and division. For instance, an expression like \((2x + y - 3z)^2\) is more than just a group of terms—

• it represents a mathematical statement where every component (like \(2x\), \(y\), and \(-3z\)) can have potential values.

When handling algebraic expressions like this one, especially in the context of expansion,

• it's essential to fully understand each term and its role.

With polynomials, these terms come together to form a whole, expanded form. Each individual variable and coefficient has its significance, contributing to the expression's overall value in different contexts.