Problem 68 - Algebra A Combined Function

Question

Multiply. \((7 y+2)^{2}\)

Step-by-Step Solution

Answer

The expanded form of \((7y + 2)^2\) is \(49y^2 + 28y + 4\).

Step 1: Understanding the Problem

The problem requires us to multiply \((7y + 2)^2\), which means we need to find the square of the binomial \(7y + 2\). This involves multiplying the binomial by itself.

Step 2: Applying the Binomial Theorem

We can use the formula for squaring a binomial, which is \((a + b)^2 = a^2 + 2ab + b^2\). Here, \(a = 7y\) and \(b = 2\).

Step 3: Calculating the First Term

Calculate \(a^2\) which is \((7y)^2 = 49y^2\).

Step 4: Calculating the Middle Term

Calculate \(2ab\), which is \(2 \times 7y \times 2 = 28y \times 2 = 28y\).

Step 5: Calculating the Last Term

Calculate \(b^2\), which is \(2^2 = 4\).

Step 6: Combining the Terms

Combine all the terms we calculated: \(49y^2 + 28y + 4\). This is the expanded form of \((7y + 2)^2\).

Key Concepts

Polynomial Multiplication

When dealing with polynomial multiplication, we extend the process of multiplying numbers to expressions that contain variables. In our example, we are multiplying the binomial \((7y + 2)\) by itself to find its square. This means that every term in the first binomial is multiplied by every term in the second binomial.
Here are some points to remember:

• Each term in the first polynomial must be multiplied by each term in the second polynomial.
• Combine like terms after multiplication to simplify your expression.
• Pay attention to signs (positive and negative) to ensure accuracy.

This process not only reinforces basic multiplication skills but also sets the stage for more complex algebraic manipulation.

Binomial Theorem

The binomial theorem provides a formula for expanding expressions that are raised to a power, like our example \((7y + 2)^2\). It simplifies the process of polynomial multiplication by using a standardized formula. In essence, the binomial theorem tells us that:

• The expanded form of \((a + b)^n\) is a sum involving terms of the form \(C(n,k) \, a^{n-k} \, b^k\), where \(C(n,k)\) is a binomial coefficient.
• For squaring a binomial \((a+b)^2\), the formula is \(a^2 + 2ab + b^2\).
• This formula helps in calculating the expanded form swiftly and without error.

Knowing this theorem allows you to expand binomials of any power confidently and accurately.

Algebraic Expressions

Algebraic expressions involve combining numbers and variables according to the algebraic operations you've learned, like addition, subtraction, multiplication, and division. An important skill is to recognize and manipulate these expressions effectively, especially when expanding or factoring them.
Key concepts include:

• Terms: A term is a single mathematical expression. In the expression \(49y^2 + 28y + 4\), each element separated by a plus or minus sign is a term.
• Coefficients: These are numbers that multiply a variable. In \(49y^2\), 49 is the coefficient.
• Like terms: Terms that have the same variable raised to the same power, which can be combined to simplify expressions.

By understanding algebraic expressions at a deeper level, you can effectively tackle polynomial multiplication and apply the binomial theorem with precision.