Question

Which of the following is equivalent to \(\left(12 x^2+4 x+5 y\right)+\left(3 x^2-2 x+\right.\) \(3 y) ?\) A) \(2 x^2-2 x+8 y\) B) \(2 x^2+15 x+8 y\) C) \(15 x^2-2 x+8 y\) D) \(15 x^2+2 x+8 y\)

Short answer

The short answer is: D) \(15x^2 + 2x + 8y\)

Step-by-step solution

Combine like terms with x^2

Add the coefficients of each x^2 term: \(12x^2 + 3x^2 = 15x^2\).

Combine like terms with x

Add the coefficients of each x term: \(4x - 2x = 2x\).

Combine like terms with y

Add the coefficients of each y term: \(5y + 3y = 8y\).

Write the simplified expression

Combine the terms from Steps 1, 2, and 3: \(15x^2 + 2x + 8y\).

Compare the simplified expression to the given choices

Our simplified expression is \(15x^2 + 2x + 8y\), which matches choice D. Therefore, the correct answer is \(D) \ 15x^2 + 2x + 8y\).

Key concepts

Combining Like Terms

In mathematics, especially when working with polynomials, it's important to know how to combine like terms. Like terms are terms in an algebraic expression that have the same variable raised to the same power. For example, in the expression \( 12x^2 + 4x + 3x^2 \), the like terms are \( 12x^2 \) and \( 3x^2 \) because they both have the variable \( x \) raised to the power of 2.

• To combine these, you simply add or subtract their coefficients, which are the numerical parts in front of the variables.
• For instance, adding \( 12x^2 \) and \( 3x^2 \) gives you \( 15x^2 \).

Combining like terms simplifies the expression and makes it easier to work with. Always remember to keep the variable part the same and to only work with the coefficients.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations, like addition, subtraction, multiplication, and division. For example, the expression \( 12x^2 + 4x + 5y \) is an algebraic expression with three terms.

• Each term in an algebraic expression can be a variable, a number, or a product of variables and numbers.
• The operation signs (plus or minus) separate different terms in the expression.

Understanding how to structure and dissect algebraic expressions is crucial for manipulating them. The goal is often to simplify these expressions as much as possible while still maintaining equivalent value, which involves careful attention to combining like terms and keeping track of variable coefficients.

Simplifying Polynomials

Simplifying polynomials is a fundamental skill in algebra. A polynomial is an algebraic expression consisting of multiple terms. By simplifying polynomials, we mean combining like terms and rewriting the expression in a more concise form.

• Identify and group like terms: Terms with the same variable factor and exponent are combined first. For example, \( 12x^2 \) and \( 3x^2 \) would be combined as \( 15x^2 \).
• Repeat this process with each type of term: Do the same for terms with the variable \( x \) and \( y \), as seen in the expression \( 4x - 2x \).

Continuously simplifying polynomials helps to reduce complexity, making equations easier to solve or evaluate. In our exercise, this results in the expression \( 15x^2 + 2x + 8y \). This process not only clarifies answers but often reveals patterns or insights, useful in broader mathematical problem-solving.