Question

What is the sum of \(3 x^{3}+5 x-3\) and \(2 x^{2}-4 x+6\) (A) \(x^{2}-9 x+9\) (B) \(2 x^{3}+4 x^{2}-x\) (C) \(3 x^{3}-x^{2}+2 x+3\) (D) \(3 x^{3}+2 x^{2}+x+3\)

Short answer

The correct answer is (D) \(3x^3 + 2x^2 + x + 3\).

Step-by-step solution

Understand the Problem

We need to find the sum of two polynomials: \(3x^3 + 5x - 3\) and \(2x^2 - 4x + 6\). Each polynomial has terms with different powers of \(x\).

Align Like Terms

Write both polynomials in a standard form, aligning like terms. The polynomial terms aligned by their degree look like this:- \(3x^3 + 0x^2 + 5x - 3\)- \(0x^3 + 2x^2 - 4x + 6\)

Add the Polynomials

Add the coefficients of like terms from both polynomials. This gives:- \( (3 + 0)x^3 + (0 + 2)x^2 + (5 - 4)x + (-3 + 6) \).

Simplify the Result

Simplify the expression from Step 3:- Combine like terms: \(3x^3 + 2x^2 + 1x + 3\).

Match with Given Options

Now, compare the simplified polynomial \(3x^3 + 2x^2 + x + 3\) to the given options:- Option (D): \(3x^3 + 2x^2 + x + 3\) matches our result.

Key concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. It is a unifying thread of almost all of mathematics and involves working with mathematical expressions using a variety of operations.
When dealing with polynomials, algebra helps us organize and simplify terms efficiently to find solutions. Algebra uses symbols to represent numbers in expressions and equations. These symbols allow us to manipulate the expressions to solve problems in a general form.
Important components of algebra include:

• Variables: Symbols used to represent numbers or values, often denoted by letters like \(x\) or \(y\).
• Coefficients: Numerical factors that multiply variables, such as 3 in \(3x\).
• Operations: Mathematical procedures like addition, subtraction, multiplication, and division.

Understanding these concepts is crucial when working with polynomial addition, as it requires adding and combining like terms—terms that have the same variables raised to the same power.

Polynomial Expressions

Polynomial expressions are algebraic expressions that consist of variables and coefficients, involving only the operations of addition, subtraction, and non-negative integer exponents of variables. Polynomials are generally expressed in descending order of the degrees of their terms.
A polynomial like \(3x^3 + 5x - 3\) has three terms, and each term is made up of a coefficient and a power of \(x\). Similar to numbers, polynomials can be added, subtracted, and multiplied.
To add two polynomial expressions, we simply combine the like terms. These are terms that have the same power of \(x\). For example, when adding \(3x^3 + 5x - 3\) and \(2x^2 - 4x + 6\):

• Align the expressions by their degrees: \(3x^3 + 0x^2 + 5x - 3\) and \(0x^3 + 2x^2 - 4x + 6\).
• Combine the coefficients of the like terms resulting in the expression: \(3x^3 + 2x^2 + 1x + 3\).

This results in a simplified and organized polynomial that aids in understanding its characteristics and behavior, paving the way for further algebraic analysis.

Math Problem Solving

Math problem solving is an essential skill that involves finding solutions to mathematical challenges by logically working through steps. It promotes critical thinking as one breaks down complex problems into more manageable parts.
Effective problem solving in math involves a few vital steps:

• Understanding the problem: Clearly define what is needed. In our example, we need to find the sum of two polynomials.
• Planning how to solve it: Decide on the steps, such as organizing and aligning terms.
• Carrying out the plan: Execute by performing operations like addition and simplification.
• Reviewing the result: Check the solution against the expected outcomes.

The systematic approach used to add the polynomials \(3x^3 + 5x - 3\) and \(2x^2 - 4x + 6\) involves turning abstract problems into concrete solutions, improving understanding and removing ambiguity. Solving problems this way not only finds an answer but deepens comprehension, vital for tackling various mathematical challenges effectively.