Question

Multiply: \((4 a-b)^{2}\)

Short answer

(4a - b)^2 = 16a^2 - 8ab + b^2

Step-by-step solution

- Identify the Formula

Recognize that \( (4a - b)^2 \) is a binomial squared. Use the formula \[ (x - y)^2 = x^2 - 2xy + y^2 \] where \ x = 4a \ and \ y = b. \

- Square the First Term

Square the first term \( 4a \) to get \[ (4a)^2 = 16a^2 \]

- Multiply the Inner Terms and Double the Product

Calculate the product of the two terms \( 4a \) and \ b \ and multiply it by 2: \[ 2 \times (4a) \times b = 8ab \]

- Square the Second Term

Square the second term \( b \) to get \[ b^2 \]

- Combine the Results

Combine the results from steps 2, 3, and 4: \[ (4a - b)^2 = 16a^2 - 8ab + b^2 \]

Key concepts

Understanding Algebra

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. It's like the language of mathematics because it helps us describe relationships and solve problems. For example, in the expression \((4a-b)^2\), \(a\) and \(b\) are variables that can represent any number.

• Variables: Symbols that represent numbers.
• Expressions: Combinations of variables and numbers using operations like addition and multiplication.

Understanding algebra is key to mastering more complex math topics!

The Basics of Polynomials

Polynomials are expressions with multiple terms. Each term is a product of a constant and one or more variables raised to a power. In our example, \((4a - b)^2\), when expanded, we get a polynomial: able gory {\(16a^2 - 8ab + b^2\).

• Terms: Parts of the expression separated by addition or subtraction.
• Coefficient: The constant factor in a term (e.g., \(16\) in \(16a^2\)).
• Degree: The highest power of the variable (e.g., \(2\) in \(16a^2\)).

Polynomials are used in various topics such as solving equations, analyzing functions, and modeling real-world scenarios.

Identifying and Working with Binomials

A binomial is a polynomial with two terms. For instance, \(4a - b\) is a binomial. When binomials are squared, like \((4a - b)^2\), they follow a specific pattern. The general formula for squaring a binomial is \((x - y)^2 = x^2 - 2xy + y^2\). Applying this to our example:

• First Term Squared: \(f(4a)^2 = 16a^2\)
• Product of Terms Doubled: \(2 \times (4a) \times b = 8ab\)
• Second Term Squared: \(b^2\)

Combining these results gives us the expanded polynomial.

Squaring Binomials Step-by-Step

When you square a binomial, you follow a specific set of steps:
Recognize the binomial form \((4a - b)^2\).
Use the formula: \((x - y)^2 = x^2 - 2xy + y^2\).
Identify: \(x = 4a\) and \(y = b\).

• Step 1: \(f(4a) = 16a^2\) (Square the first term).
• Step 2: Multiply the two terms and double the result \(2 \times (4a) \times b = 8ab\).
• Step 3: \(f(b) = b^2\). (Square the second term).
• Step 4: Combine all the results: \(16a^2 - 8ab + b^2\).

This method ensures you never miss a step!