Problem 32 - Algebra A Combined Function

Question

Subtract using a vertical format. $$ \begin{array}{r} 7 a^{2}-9 a+6 \\ -\left(11 a^{2}-4 a+2\right) \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The result is \(-4a^{2} - 5a + 4\).

Step 1: Write the Polynomials Vertically

Before performing the subtraction, write the two polynomials vertically, aligning like terms:\[\begin{array}{r}7a^{2} - 9a + 6 \- (11a^{2} - 4a + 2) \\hline\end{array}\]

Step 2: Distribute Negative Sign

Distribute the negative sign across the second polynomial. Change the signs of each term in the second polynomial:\[\begin{array}{r}7a^{2} - 9a + 6 \-11a^{2} + 4a - 2 \\hline\end{array}\]

Step 3: Subtract Like Terms

Subtract the corresponding like terms in the polynomials:- Subtract the quadratic terms: \(7a^{2} - 11a^{2} = -4a^{2}\).- Subtract the linear terms: \(-9a + 4a = -5a\).- Subtract the constant terms: \(6 - 2 = 4\).

Step 4: Write the Resultant Polynomial

Combine the results from Step 3 into a single polynomial:\[-4a^{2} - 5a + 4\].

Key Concepts

Vertical Subtraction

Vertical subtraction is a fantastic technique to simplify the subtraction of polynomials. Handling polynomials this way makes it more clear which terms you are subtracting from one another. To start off, line up the polynomials in a column format. Make sure that you align each term with its counterpart in the other polynomial. For example, place the quadratic terms on top of each other, like terms with like terms, etc. Once they are all set up, this arrangement will help visually organize your work:

• Higher degree terms (like quadratic terms) come first.
• Linear terms are next.
• Finally, constant terms are at the bottom.

Keeping everything in neat columns makes you less likely to make errors when you start subtracting. By focusing on one group of terms at a time, you will be able to clearly see how each part of the polynomial should be handled.

Like Terms

When subtracting polynomials, recognizing and understanding 'like terms' is fundamental. Like terms are terms within a polynomial that have the same variables raised to the same powers. This means the coefficients of these terms are the only parts that differ.

• For instance, in the polynomials \(7a^2 - 9a + 6\) and \(11a^2 - 4a + 2\), 'like terms' are \(7a^2\) with \(11a^2\), \(-9a\) with \(-4a\), and \(6\) with \(2\).

When subtracting these, only the coefficients of each like term are operated on. This is crucial because mixing unlike terms would lead to incorrect results. In this context, understanding the concept of like terms ensures precision throughout the process.

Distributing Negative Sign

Before commencing with subtraction, it's necessary to distribute the negative sign across the entire second polynomial. This is a crucial step since failing to do so can result in significant mistakes. The negative sign affects each term in the second polynomial:

• Each term’s sign changes, turning positives to negatives and negatives to positives.

For our exercise: the polynomial \(11a^2 - 4a + 2\) becomes \(-11a^2 + 4a - 2\). Once the signs have been adjusted, you can think of simply adding these adjusted terms to the first polynomial. This makes the subtraction process clearer and easier to handle, since now you’re effectively just dealing with addition operations across each column of like terms.