Problem 10 - Algebra 1: Concepts and Skills

Question

Tell whether the statement is true or false. If the statement is false, rewrite the right-hand side to make the statement true. $$ (3+2 y)^{2}=9+12 y+4 y^{2} $$

Step-by-Step Solution

Answer

The statement \( (3+2 y)^{2}=9+12 y+4 y^{2} \) is true, thus there is no need to rewrite the right-hand side.

Step 1: Understanding the Expression

The problem statement provides an equation based on a binomial theorem, which has a power of 2. You are required to verify the provided right-hand side by expanding the left-hand side. The expression to verify is \( (3+2y)^{2}=9+12y+4y^{2} \) .

Step 2: Expand the left-hand side

Expanding \( (3+2y)^{2} \) using the binomial theorem, it will be \( (3+2y)^{2} = 3^2+2(3)(2y)+(2y)^2 = 9+12y+4y^{2} \).

Step 3: Compare the results and conclude

After comparing \( 9+12y+4y^{2} \) (the expanded left-hand side) and \( 9+12y+4y^{2} \) (the given right-hand side), we can see they are identical. Therefore, the given problem statement is true.

Key Concepts

Algebraic expansion

Algebraic expansion is a technique used to simplify expressions, especially when dealing with binomials. Binomials are algebraic expressions containing two terms, usually connected by a plus or minus sign. Expanding these expressions helps convert them into a form that is easier to manage, especially when looking to verify equations or identify polynomial components.

In our exercise, we dealt with expanding \((3+2y)^2\) using the binomial theorem. The binomial theorem is designed to expand powers of binomials and states that for any positive integer \(n\), \((a+b)^n\) can be expanded into a sum involving terms of the form \(C(n,k)a^{n-k}b^k\). Here, \(C(n,k)\) is the binomial coefficient. However, for a squared binomial, like \((a+b)^2\), a simple pattern emerges:

\[(a+b)^2 = a^2 + 2ab + b^2\].

This pattern allows us to directly calculate the expanded form efficiently, just like we did for \((3 + 2y)^2 = 3^2 + 2 \cdot 3 \cdot 2y + (2y)^2\), which resulted in \(9 + 12y + 4y^2\).

Understanding and applying algebraic expansion is crucial in simplifying complex equations into easily verifiable or solvable forms.

Polynomial

A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. An essential feature of polynomials is their "degree," which signifies the highest exponent of the variable in the expression.

In this exercise, the polynomial on the right-hand side of the equation \(9 + 12y + 4y^2\) is a typical quadratic polynomial. Its terms are ordered from the highest degree to the lowest: the highest degree here is 2, represented by \(4y^2\). This quadratic form is characteristic because of its inclusion of a squared term, a linear \(y\) term, and a constant.

Polynomials like these are valuable in various mathematical computations, from basic algebraic manipulation to solving equations and modeling real-world phenomena, such as projectile motion or economic predictions. Recognizing and understanding the structure of a polynomial helps in identifying solutions or analyzing behavior in corresponding algebraic functions.

Verification of equations

Verification of equations involves checking whether two expressions are equivalent, often by manipulating one side to see if it matches the other. This ensures that an equation holds true in all situations and is a fundamental aspect of algebraic problem-solving.

In the context of the given problem, verifying the equation \((3+2y)^2 = 9 + 12y + 4y^2\) was performed by expanding the left-hand side. By using the algebraic expansion method, we obtained the expanded form that matched the right-hand side: \(9 + 12y + 4y^2\).

This match confirmed the truth of the statement. Verification can often reveal mistakes or misconceptions in algebraic formulation and is a reliable way to ensure the accuracy of solutions in mathematics.