Problem 40 - Algebra 1: Concepts and Skills

Question

Use a horizontal format to add or subtract. $$ \left(-3 a^{2}+5\right)+\left(-a^{2}+4 a-6\right) $$

Step-by-Step Solution

Answer

-4a^2 + 4a - 1

Step 1: Identify Like Terms

There are three like terms in the given polynomials: \( -3a^{2} \) and \( -a^{2} \), \( 5 \) and \( -6 \), and \( 4a \), which stands alone.

Step 2: Add Like Terms

Combining the like terms gives the following results: \( -3a^{2} + -a^{2} = -4a^{2} \), \( 5 + -6 = -1 \), and \( 4a = 4a \).

Step 3: Write the Final Result

Combine these results to write the final answer: \( -4a^{2} + 4a - 1 \). Ensure that the terms are written in descending order according to their degree (highest power of a).

Key Concepts

Combining Like Terms

When it comes to polynomial addition and subtraction, the cornerstone concept is the idea of combining like terms. This technique involves summing or subtracting coefficients of terms that have the same variable raised to the same power. In simpler terms, look for terms that have the same letter part and exponent.

For example, consider the polynomial addition \( -3a^{2} + 5 \)+\( -a^{2} + 4a - 6 \). Here, the like terms are those with the same variable and degree, such as the terms \( -3a^{2} \) and \( -a^{2} \), which both involve \( a^{2} \) and can thus be added together.

Per the exercise provided, the process is straightforward:

• Identify the like terms: \( -3a^{2} \) and \( -a^{2} \) (both are \( a^{2} \) terms), \( 5 \) and \( -6 \) (constant terms), and \( 4a \) stands alone since there are no other \( a \) terms with the same degree.
• Add or subtract the coefficients of these like terms: \( -3a^{2} -a^{2} = -4a^{2} \) and \( 5 - 6 = -1 \).
• The term \( 4a \) remains unchanged as it has no like term to combine with.

Understanding and applying the concept of combining like terms simplifies polynomial arithmetic significantly, making it a foundational skill for algebra.

Polynomial Arithmetic

Moving beyond the basics, polynomial arithmetic involves operations like addition, subtraction, multiplication, and division with polynomials. Focusing on addition and subtraction, the key steps align with arithmetic you already know but with a twist: you must respect the variable parts of the terms.

In polynomial addition and subtraction, as seen in the given exercise, you perform the following actions:

• First, scan the expression for like terms, similar to grouping numbers in basic arithmetic. Like terms are those terms that have the exact variable part (including the degree).
• Then, combine the coefficients of these like terms through addition or subtraction.
• Finally, keep all the variable parts unchanged while preserving the original signs of the terms unless they are affected by subtraction.

When subtracting polynomials, a helpful tip is to distribute the negative sign (if present) across the polynomial being subtracted before combining like terms. This avoids sign errors and ensures accuracy. Once you have combined the like terms, you are now adept at the 'arithmetic' part of polynomial arithmetic, a crucial skill for advancing in algebra.

Descending Order of Degrees

The last step in the process of polynomial addition and subtraction, and a commonly enforced convention in algebra, is arranging the terms in descending order of their degrees. The degree of a term with a variable is the exponent on the variable. When you write a polynomial, you'll typically list the terms starting with the highest degree and moving down to the lowest.

For instance, after combining like terms in our example as \( -4a^{2} + 4a - 1 \), we already have the polynomial in a descending order since \( a^{2} \) is of degree 2, \( a \) is of degree 1, and the constant \( -1 \) is of degree 0 (since \( a^0 = 1 \)).

This ordering makes it easy to compare polynomials and is essential when you perform other operations, such as long division or factoring. It also provides a standardized way of expressing polynomials, making mathematical communication clearer. Thus, the practice of writing polynomials in descending order of degrees is not just about maintaining an aesthetic standard; it's about mathematical clarity and precision.