Problem 57 - Beginning and Intermediate Algebra

Question

Subtract. $$\begin{array}{r} 2 b^{2}-7 b+4 \\ -3 b^{2}+5 b-3 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

\(5b^2 - 12b + 7\)

Step 1: Identify the corresponding terms for subtraction

Identify the terms with the same exponents in each polynomial. In this case, we have \(2b^2\), \(-7b\), and \(4\) in the first polynomial, and \(-3b^2\), \(5b\), and \(-3\) in the second polynomial.

Step 2: Subtract the terms with the same exponents

Now, we will subtract the terms with the same exponents: \[(2b^2) - (-3b^2), \quad (-7b) - (5b), \quad 4 - (-3)\]

Step 3: Simplify the expressions

Next, we will simplify each expression we found in step 2: \[(2b^2 +3b^2), \quad (-7b - 5b), \quad (4 + 3)\]

Step 4: Write the result as a single polynomial

Finally, write down the simplified expressions as a single polynomial: \[5b^2 - 12b + 7\] The result of subtracting the second polynomial from the first is: \[5b^2 - 12b + 7\]

Key Concepts

Polynomial Expressions

Polynomial expressions are foundational elements in algebra. They are combinations of variables and coefficients. In a polynomial, terms are formed by variables raised to non-negative integer exponents, and each term consists of a coefficient (a numerical factor).
A polynomial could look like this:

• \(2b^2\)
• \(-7b\)
• \(4\)

Each term has a specific component: the coefficient and the variable's power or degree. In the example, \(b^2\) indicates that the variable \(b\) is squared in the first term. Polynomial expressions are a crucial aspect of working with equations in algebra. They allow for algebraic operations like addition, subtraction, multiplication, and division between them.

Algebraic Operations

Algebraic operations involve the basic mathematical operations of addition, subtraction, multiplication, and division performed on algebraic expressions. When working with polynomials, these operations must be applied with care to ensure accuracy.
For subtraction, as in the given exercise, you must subtract each corresponding term's coefficients, considering the sign of each term. This involves the application of:

• Distributing the subtraction sign across each term of the polynomial being subtracted.
• Combining or cancelling out terms with matching powers or exponents.

These operations help in transforming and simplifying polynomial expressions to find solutions to equations. In our example, we subtract corresponding terms from each polynomial.

Like Terms

Like terms in polynomials are terms that have the same variable raised to the same power. Identifying like terms is key to performing operations like addition or subtraction.
When you see expressions like \(2b^2\) and \(-3b^2\), these are like terms because both have the variable \(b\) raised to the power of 2. On the other hand, a term like \(-7b\) and the constant \(4\) do not have like terms in common, as they relate to different variables and powers respectively.

• Ensure each term is aligned with its like counter-part from the other polynomial.
• Only add or subtract terms if they are like terms.

By focusing on like terms, you ensure the mathematical integrity of your polynomial expressions, maintaining equality and correctness throughout your operations.

Simplifying Polynomials

Simplifying polynomials entails reducing the expression to its simplest form by performing all possible algebraic operations and combining like terms.
Starting from subtraction as in our exercise, it is vital to:

• Perform operations such as addition or subtraction accurately across like terms.
• Combine remaining terms to write the polynomial in the standard form, organized by descending powers or exponents.

In our exercise, after performing subtraction on the like terms, we combined them into a single expression. Thus,
\[5b^2 - 12b + 7\]
represents the simplified form of the subtracted polynomials, with all terms accounted for and properly combined. This step is critical for making further operations or analysis on the polynomial easier and more straightforward.