Problem 33 - Algebra 1: Concepts and Skills

Question

Use a vertical format to add or subtract. $$ \left(12 x^{3}+x^{2}\right)-\left(18 x^{3}-3 x^{2}+6\right) $$

Step-by-Step Solution

Answer

The subtracted polynomial is \( -6x^{3} + 4x^{2} - 6 \)

Step 1: Rewrite the Expression

Rewrite the subtraction as an addition of the opposite: \( 12x^{3} + x^{2} + (-18x^{3} + 3x^{2} - 6) \).

Step 2: Rearrange the Terms

Rearrange the polynomial such that similar terms are positioned underneath each other; in this case, the terms with \( x^{3} \), \( x^{2} \), and the numeral. \[\begin{align*} 12x^{3} & + x^{2} & & \ -18x^{3} & + 3x^{2} & - 6 \ \end{align*} \]

Step 3: Subtract the Like Terms

Subtract the matching terms from each other to get the final polynomial. \[\begin{align*} (12x^{3} - 18x^{3}) & + (x^{2} + 3x^{2}) & + (-6) \ = -6x^{3} & +4x^{2} & -6 \ \end{align*} \]

Key Concepts

Understanding Like Terms

When dealing with polynomials, identifying like terms is crucial. Like terms are those that share the same variable raised to the same power. For example, in the expression \(12x^3 + x^2 - 18x^3 + 3x^2 - 6\), the terms \(12x^3\) and \(-18x^3\) are like terms because they both include \(x^3\). Similarly, \(x^2\) and \(3x^2\) are like terms as they both include \(x^2\).

• To simplify an expression, group like terms together.
• Especially in subtraction, pay close attention to signs.
• Combine the coefficients of like terms for simplification.

Combining like terms streamlines solving polynomial equations, enabling straightforward simplification and calculation.

Using Vertical Format

The vertical format is a very helpful and visual way to handle polynomial subtraction or addition. Much like traditional arithmetic, this method involves writing each polynomial vertically, aligning the like terms.

• Write both polynomials one under the other.
• Make sure to align terms with the same degree, such as \(x^3\) with \(x^3\), \(x^2\) with \(x^2\).
• Fill in any missing terms with zeros to maintain consistency, especially if one polynomial is missing a certain degree term.

This method simplifies operations by visually organizing them, which helps to ensure that no terms are overlooked during the operation.

Simplifying with Polynomial Addition

To simplify expressions correctly, you often rewrite subtraction problems as addition by adding the opposite. For instance, in the given problem, \(12x^3 + x^2 - (18x^3 - 3x^2 + 6)\) is rewritten as \(12x^3 + x^2 + (-18x^3 + 3x^2 - 6)\). This converts the subtraction into an addition problem, making it easier to simply add terms.

• Adding polynomials involves combining like terms together, and each set of terms is added separately.
• Pay attention to negative signs when dealing with subtraction, as it changes the signs of the terms inside the parentheses.
• By organizing the polynomials vertically, one can easily add each set either row by row or column by column, ensuring a clear path to the solution.

Polynomial addition can simplify complex expressions into manageable form, making calculations more reliable and less prone to error.