Problem 63 - Algebra A Combined Function

Question

Add or subtract as indicated. $$ \left(11 r^{2} s+16 r s-3-2 r^{2} s^{2}\right)-\left(3 s r^{2}+5-9 r^{2} s^{2}\right) $$

Step-by-Step Solution

Answer

\( 7r^2 s^2 + 8r^2 s + 16rs - 8 \)

Step 1: Distribute the Negative Sign

The expression is given with a subtraction, so we need to distribute the minus sign over the second expression: \( (11r^2 s + 16rs - 3 - 2r^2 s^2) - (3sr^2 + 5 - 9r^2 s^2) \)Becomes:\( 11r^2 s + 16rs - 3 - 2r^2 s^2 - 3sr^2 - 5 + 9r^2 s^2 \)

Step 2: Combine Like Terms

Now, combine the like terms: - For \(r^2 s\), combine \(11r^2 s - 3sr^2\)- For \(rs\), it remains \(16rs\), as there is no like term in the second group.- For the constants, combine \(-3 - 5\)- For \(r^2 s^2\), combine \(-2r^2 s^2 + 9r^2 s^2\).This simplifies to: \( (11r^2 s - 3r^2 s) + 16rs + (-3 - 5) + (-2r^2 s^2 + 9r^2 s^2) \).

Step 3: Simplify Each Group of Like Terms

Now perform the arithmetic in each group:- For \(r^2 s\): \(11r^2 s - 3r^2 s = 8r^2 s\)- The \(rs\) term remains \(16rs\).- The constants: \(-3 - 5 = -8\)- For \(r^2 s^2\): \(-2r^2 s^2 + 9r^2 s^2 = 7r^2 s^2\) Combine these to get the final expression:\( 7r^2 s^2 + 8r^2 s + 16rs - 8 \).

Key Concepts

Like Terms

Understanding like terms is essential when dealing with polynomials, especially in addition and subtraction. Like terms refer to terms that have the exact same variable factors raised to the same power. These terms can be combined by their coefficients because they contain the same variable part.

• For example, in the expression \(11r^2s + 16rs\), the terms \(11r^2s\) and \(16rs\) are not like terms. This is because they do not have the same variable factors; one includes \(r^2\) whereas the other only has \(r\).
• However, within the expressions \(11r^2s\) and \(3sr^2\), they are like terms. Both involve \(r^2s\), only the coefficients differ, allowing them to be combined.

Noticing the structure of terms, including their coefficients and variable parts, helps in identifying like terms and simplifying expressions correctly.

Combining Like Terms

After identifying like terms in a polynomial expression, combining them is the next step. This means summing or subtracting their coefficients while keeping the variable part identical.

• In the given problem, terms involving \(r^2s\) are combined: \(11r^2s - 3r^2s = 8r^2s\). The coefficients \(11\) and \(-3\) are combined to get \(8\).
• For the constants in the expression: \(-3 - 5 = -8\). Here, we simply add the numbers since they don't involve any variables.

By ensuring that only like terms are combined, you maintain the integrity of the expressions and correctly simplify the polynomial.

Distribute Negative Sign

Distributing a negative sign is crucial when subtracting polynomials. When an expression has subtraction, the minus sign in front of a bracketed term affects every term inside the brackets, effectively changing their signs.

• For example, if you have \((3sr^2 + 5 - 9r^2s^2)\), preceded by a minus sign, distribute it to each term inside. This turns \(3sr^2\) into \(-3sr^2\), \(5\) into \(-5\), and \(-9r^2s^2\) into \(+9r^2s^2\).
• Think of the minus sign as multiplying every term in the expression by \(-1\), flipping their signs. This distribution is vital before combining like terms, ensuring the arithmetic is correct.

Properly handling the negative sign helps avoid mistakes in polynomial subtraction, ensuring accurate results.