Problem 25 - A Graphical Approach to College Algebra

Question

Find each sum or difference. $$\left(8 p^{2}-5 p\right)-\left(3 p^{2}-2 p+4\right)$$

Step-by-Step Solution

Answer

The solution is \(5p^2 - 3p - 4\).

Step 1: Distribute the Negative Sign

First, distribute the negative sign into the second polynomial, changing the operation from subtraction to addition. This means you change the signs of each term in the second polynomial:\(- (3p^2 - 2p + 4) = -3p^2 + 2p - 4\).

Step 2: Re-write the Expression

Re-write the expression with the distributed terms from Step 1:\( (8p^2 - 5p) + (-3p^2 + 2p - 4) \).

Step 3: Combine Like Terms

Combine the terms with the same powers of \(p\). This is done by grouping them together and adding/subtracting:\[(8p^2 - 3p^2) + (-5p + 2p) + (-4) = 5p^2 - 3p - 4.\]

Step 4: Simplify the Result

Simplify the expression obtained by performing the arithmetic in the combined terms:- \(8p^2 - 3p^2 = 5p^2\)- \(-5p + 2p = -3p\)- The constant term is \(-4\).Thus, the simplified expression is: \(5p^2 - 3p - 4\).

Key Concepts

Distributive Property

The Distributive Property is a basic principle in algebra that is often used to simplify expressions. It's like a rule for distributing a multiplier across terms inside parentheses. When subtracting polynomials, you’ll need to apply this property to remove parentheses by combining terms. Here’s how it works in our example. We have \((8p^2 - 5p) - (3p^2 - 2p + 4)\). Notice the subtraction before the second polynomial. The distributive property tells us to multiply each term in the second polynomial by -1. This changes:

• \(- (3p^2)\) becomes \(-3p^2\)
• \(- (-2p)\) turns into \(+2p\)
• \(- (+4)\) changes to \(-4\)

With these new terms, we rewrite the expression to make it easier to work with: \((8p^2 - 5p) + (-3p^2 + 2p - 4)\). This step is essential before moving on to combine like terms.

Combining Like Terms

Combining like terms is the action of adding or subtracting coefficients of terms that have the same variable raised to the same power. Essentially, it's about grouping similar pieces to simplify the expression. Looking at our new expression from the previous step: \((8p^2 - 5p) + (-3p^2 + 2p - 4)\):

• The terms \(8p^2\) and \(-3p^2\) are like terms because they both have \(p^2\). We combine them: \(8p^2 - 3p^2 = 5p^2\).
• The terms \(-5p\) and \(+2p\) are also like terms because they both contain \(p\). We combine them: \(-5p + 2p = -3p\).
• The number \(-4\) is a constant and stands alone in this equation since no other constant is present.

Combining like terms is crucial as it transforms a complex expression into something much simpler, laying groundwork for the final expression.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is the process of cleaning up the expression to its most concise and reduced form. This involves both distributing and combining like terms, which we’ve already done. Starting with our expression after combining like terms: \(5p^2 - 3p - 4\).

• The term \(5p^2\) represents all the \(p^2\) components within our expression.
• The term \(-3p\) represents all the \(p\) components.
• Finally, the constant term \(-4\) is the standalone number that completes the expression.

The final simplified expression remains \(5p^2 - 3p - 4\). This form is the simplest version of the original problem, ensuring it is as clear and straightforward as possible. Simplifying expressions in this way aids in easier interpretation and use in further algebraic operations or evaluations.