Question

$$\left(4 y+2 y^{2}\right)-\left(3 y-6 b+4 y^{2}+5\right)$$

Short answer

The simplified form of the expression is \(2 y^2 + y + 6b + 5\).

Step-by-step solution

Distribute the Negative Sign

Apply the negative sign to each term of the second parenthesis, changing the sign of each term within the parenthesis before subtracting from the first.

Combine Like Terms

After distributing the negative sign, combine like terms by adding or subtracting the coefficients of the terms with the same variables.

Simplify the Expression

Simplify the expression by executing the addition and subtraction of the coefficients found in Step 2 to find the final result.

Key concepts

Distributive Property

Understanding the distributive property is fundamental to simplifying algebraic expressions. It allows you to multiply a single term by each term within a parenthesis. For example, if you have an expression like \( a(b + c) \), you can distribute the 'a' across the terms inside the parenthesis to get \( ab + ac \).

In the exercise provided, the distributive property is used slightly differently. When you have a negative sign before a parenthesis, like \( -(d + e - f) \), you must distribute this negative sign to all the terms inside the parentheses, changing each term's sign. It transforms the expression to \( -d - e + f \). This is the first step taken in the solution provided. It changes the sign of all terms in the second parentheses, allowing us to then combine like terms effectively.

Combining Like Terms

Once you've distributed the negative sign over the terms, the next step in the simplification process is to combine like terms. Like terms are terms that have the exact same variable raised to the same power. They're like 'apples to apples' in math and can be easily added or subtracted.

For example, if you have \( 2x \) and \( 3x \), they're like terms because they both contain the variable 'x' raised to the same power. So, \( 2x + 3x = 5x \). However, if you have \( x \) and \( y \) or \( x^2 \) and \( x^3 \), they are not like terms and can't be combined in this way.

In our exercise, after applying the negative sign, we have to identify terms with the same variables and their exponents, such as terms with \( y \) and terms with \( y^2 \), and then combine them by adding or subtracting their numerical coefficients.

Simplify Algebraic Expressions

Finally, to simplify an algebraic expression means to make it as straightforward as possible. This involves performing the operation of combining like terms and reducing the expression to its most concise form. It's like tidying up a room - you're organizing and getting rid of unnecessary clutter.

For the given exercise, once you've distributed the negative and combined like terms, you then add or subtract the coefficients as applicable. This step finalizes the simplification process, leading to an expression that is easier to read and work with. It's crucial to follow the 'order of operations' when simplifying, ensuring you perform operations within parentheses first, then multiplication and division, and finally addition and subtraction.

The end result should be a clear, concise algebraic statement that represents the same quantity as the original expression but in a more streamlined manner.