Problem 67 - Elementary Algebra

Question

Are the following factored correctly? Check by multiplying. $$ 3 x 3-5 x 6=x 3(3-x 2) $$

Step-by-Step Solution

Answer

The expression is not factored correctly.

Step 1: Observing the Factored Form

The given expression is \(3x^3 - 5x^6 = x^3(3 - x^2)\). We see that the expression is alleged to be factored into \(x^3(3 - x^2)\).

Step 2: Multiply the Factored Form

To verify if the factored form is correct, expand \(x^3(3 - x^2)\) by distributing \(x^3\) to each term inside the parentheses:- First multiply: \(x^3 \times 3 = 3x^3\)- Then multiply: \(-x^3 \times x^2 = - x^5\)Combining these results gives: \(3x^3 - x^5\).

Step 3: Compare the Expanded Form with Original Expression

The expanded form \(3x^3 - x^5\) does not match the original expression \(3x^3 - 5x^6\). Therefore, the provided factored expression is incorrect.

Key Concepts

Polynomials

Polynomials are fundamental expressions in algebra consisting of variables and coefficients, involving operations of addition, subtraction, and non-negative integer exponents of variables. These can range from simple forms like a single variable to more complex forms with multiple terms. In our exercise, the polynomial given is \(3x^3 - 5x^6\), which includes two terms: \(3x^3\) and \(-5x^6\). Each term is composed of a coefficient (the numbers 3 and -5) and the variable \(x\), raised to the power of 3 and 6 respectively. Understanding the structure of polynomials is crucial because they are the building blocks of many algebraic operations, including factoring and expanding expressions. Being comfortable with identifying terms, degrees (which is the highest power of the variable), and the concept of like terms will ease the process of solving polynomial-related problems.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations, which together define a specific value or set of values. In solving algebra problems, algebraic expressions can be manipulated by various algebraic techniques such as factoring, expanding, adding, or subtracting. The given problem's expression, \(3x^3 - 5x^6\), is an algebraic expression. When the exercise asks if the expression \(x^3(3 - x^2)\) correctly factors \(3x^3 - 5x^6\), it involves understanding how algebraic expressions can be split or simplified. This echoes the importance of using correct algebraic techniques to ensure the simplified expression resolves back to its original form when re-expanded. Proper handling of these expressions involves attention to mathematical rules, ensuring all parts of the expression are accounted for.

Mathematical Verification

Mathematical verification is the process of proving that a mathematical statement or solution is logically sound by employing specific methods, ensuring accuracy in problem-solving. In the given exercise, verification requires expanding the supposed factored form, \(x^3(3 - x^2)\), to see if it matches the original polynomial \(3x^3 - 5x^6\).

• Begin by multiplying each term in the factored expression by \(x^3\).
• This gives \(3x^3\) from \(x^3 \times 3\) and \(-x^5\) from \(-x^3 \times x^2\).
• Comparing \(3x^3 - x^5\) shows it does not match \(3x^3 - 5x^6\).

This comparison showcases the powerful tool of verification, where such mismatches highlight errors in the factoring process. Verification helps ensure that calculations adhere to given constraints or requirements, thereby helping students avoid errors and understand problem-solving strategies more deeply.