Problem 51 - Beginning and Intermediate Algebra

Question

Subtract. $$\left(-3 b^{4}-5 b^{2}+b+2\right)-\left(-2 b^{4}+10 b^{3}-5 b^{2}-18\right)$$

Step-by-Step Solution

Answer

The short answer is: \[ -b^4 - 10b^3 + b + 20 \]

Step 1: Rewrite the expression

Let's rewrite the given expression by changing the sign of each term in the second polynomial: \(( -3b^4 - 5b^2 + b + 2) - ( -2b^4 + 10b^3 - 5b^2 -18) \) This can be simplified as: \( -3b^4 - 5b^2 + b + 2 + 2b^4 - 10b^3 + 5b^2 +18 \)

Step 2: Combine like terms

Now let's combine the like terms (terms with the same degree of variable \(b\)): \( (-3b^4 + 2b^4) + (- 10b^3) + (-5b^2 + 5b^2) + (b) + (2 + 18) \)

Step 3: Simplify the expression

Simplify the expression by performing the operations: \( (-1b^4) + (- 10b^3) + (0b^2) + (b) + (20) \) Since "0b²" equals zero, we can remove it from the expression: \( -b^4 - 10b^3 + b + 20 \) The final result after subtracting the second polynomial from the first one is: \[ -b^4 - 10b^3 + b + 20 \]

Key Concepts

Combining Like Terms

In algebra, combining like terms is an essential skill that helps simplify expressions. **Like terms** are terms that have the same variable raised to the same power. For example, in the expression \(-3b^4 - 5b^2 + 2b^4\), the terms \(-3b^4\) and \(2b^4\) are like terms because they both involve the variable \(b^4\). When combining like terms, follow these steps:

• Identify terms with the same variable and exponent.
• Add or subtract their coefficients (the numerical part of the terms).

This process makes expressions simpler and easier to work with. In our exercise, after rewriting the expression to include all terms from both polynomials, we were able to combine \(-3b^4\) with \(2b^4\) to get \(-b^4\). Similarly, terms like \(-5b^2\) and \(5b^2\) canceled each other out because their sum is zero.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, making them easier to read and work with. This process includes combining like terms, removing unnecessary terms, and performing arithmetic operations. Here are the key steps to simplify algebraic expressions:

• First, ensure that all like terms are combined.
• Perform any arithmetic, such as addition or subtraction, to consolidate coefficients.
• Eliminate any terms that sum to zero, such as in the case of \(0b^2\), which should be removed from the expression.

In the original expression \(-3b^4 - 5b^2 + b + 2 + 2b^4 - 10b^3 + 5b^2 +18\), each step carefully reduced terms until we reached the simplified form \(-b^4 - 10b^3 + b + 20\). This final expression is more manageable and straightforward.

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operations. It can represent a single value or a complex relationship between different quantities. The original problem involves subtracting one polynomial from another, which constitutes handling an algebraic expression.Polynomials are a type of algebraic expression. They have multiple terms, each consisting of a coefficient (a number) and variables raised to a power (such as \(b^4\)). When dealing with algebraic expressions:

• Understand the structure, so you can identify separate terms.
• Use operations: addition, subtraction, multiplication, and division can change or simplify expressions.
• Leverage algebraic principles like distribution or factoring, if necessary.

In this exercise, the algebraic expression involved two polynomials, and by using the principles of subtraction and combining like terms, we simplified it to a succinct form. This ability to manipulate expressions is foundational to solving algebraic equations.