Problem 59 - Elementary Algebra

Question

\((9-4 r)-\left(5 r+7 r^{2}\right)-\left(3 r^{2}+1\right)\)

Step-by-Step Solution

Answer

8 - 9r - 10r^2

Step 1: Distribute the negative sign across the second parentheses

Distribute the negative sign across each term inside the second set of parentheses: \(9 - 4r - 5r - 7r^2\).

Step 2: Distribute the negative sign across the third parentheses

Next, distribute the negative sign across the terms inside the third set of parentheses: \(9 - 4r - 5r - 7r^2 - 3r^2 - 1\).

Step 3: Combine like terms

Now, combine all like terms (constants with constants, terms involving r with each other, and terms involving r^2 with each other): \(9 - 1 - 4r - 5r - 7r^2 - 3r^2 \)

Step 4: Simplify the expression

Simplify the expression: \(8 - 9r - 10r^2 \)

Key Concepts

Polynomial Subtraction

Polynomial subtraction involves subtracting one polynomial from another. In our exercise, we need to subtract two polynomials from the first polynomial. Remember, when subtracting polynomials, you distribute the negative sign (which is like multiplying by -1) across each term in the polynomial being subtracted. For example, in the expression \((9-4r)-\(5r+7r^2\)-\(3r^2+1\)\), you subtract the terms inside each set of parentheses from the first polynomial.

Distributive Property

The distributive property helps us to distribute a number or a sign across terms inside parentheses. In subtraction, the distributive property means that you apply the negative sign to each term inside the parentheses.
For instance, when we distribute the negative sign across the second set of parentheses (\(5r + 7r^2\)), we get:
\(9 - 4r - 5r - 7r^2\). Similarly, distributing the negative across the third parentheses (\(3r^2 + 1\)) gives:
\(9 - 4r - 5r - 7r^2 - 3r^2 - 1\).
This step is crucial to convert subtraction into addition of negative terms.

Combining Like Terms

Combining like terms involves merging terms that have the same variable and the same power. In our problem, we combine the constant terms, the terms with \(r\), and the terms with \(r^2\):

• Constants: \(9 - 1\)
• Terms with \(r\): \(-4r - 5r\)
• Terms with \(r^2\): \(-7r^2 - 3r^2\)

This results in:
\(8 - 9r - 10r^2\).
Combining like terms simplifies expressions and helps in evaluating the final solution.

Simplification

Simplification is the process of making an algebraic expression as compact and easy to understand as possible. After combining like terms, our expression \(8 - 9r - 10r^2\) is in its simplest form.
Simplification includes:

• Eliminating any unnecessary terms
• Combining any similar terms
• Making sure the expression has no parentheses

Through these processes, we transform complex expressions into their simplest and most manageable forms.