Problem 68 - Elementary and Intermediate Algebra

Question

Subtract \(-3 z^{3}-4 z+7\) from the sum of \(2 z^{2}+3 z-7\) and \(-4 z^{3}-2 z-3\)

Step-by-Step Solution

Answer

The result is \(-z^3 + 2z^2 + 5z - 17\).

Step 1: Calculate the Sum of the Polynomial Expressions

First, add the expressions \(2z^2 + 3z - 7\) and \(-4z^3 - 2z - 3\). To do this, align like terms and perform the addition:- \(2z^2 + 3z - 7\)- \(-4z^3 - 2z - 3\)Combine the terms to get \(-4z^3 + 2z^2 + (3z - 2z) + (-7 - 3)\). The resulting polynomial is \(-4z^3 + 2z^2 + z - 10\).

Step 2: Subtract the Given Polynomial

Now, subtract the polynomial \(-3z^3 - 4z + 7\) from the result of the sum we calculated in Step 1. Align like terms to facilitate subtraction:- Polynomial from Step 1: \(-4z^3 + 2z^2 + z - 10\)- Subtract: \(-3z^3 - 4z + 7\)Perform the subtraction by changing the signs of the polynomial being subtracted and then combine like terms:\[(-4z^3 - 2z^2 + z - 10) - (-3z^3 - 4z + 7) = -4z^3 + 2z^2 + z - 10 + 3z^3 + 4z - 7\]Simplify this to get \((-4z^3 + 3z^3) + 2z^2 + (z + 4z) - 10 - 7\), leading to the simplified expression \(-z^3 + 2z^2 + 5z - 17\).

Key Concepts

Polynomial Addition

Polynomial addition is the process of combining two or more polynomial expressions to form a single expression. Just like adding regular numbers, you need to add like terms together in polynomials.
When adding polynomials, let's say we have two polynomials:

• First Polynomial: \(2z^2 + 3z - 7\)
• Second Polynomial: \(-4z^3 - 2z - 3\)

To perform the addition, follow these steps:

• Align like terms vertically under each other to make the addition easier.
• Add the coefficients of the like terms, while keeping the same variables and exponents.

For the example above:

• Combine: \(-4z^3 + 2z^2 + (3z - 2z) + (-7 - 3)\)
• This simplifies to: \(-4z^3 + 2z^2 + z - 10\)

This new polynomial represents the sum of the initial polynomial expressions.

Like Terms

Identifying like terms is crucial for adding and subtracting polynomial expressions.Like terms are terms that have the same variables raised to the same powers. For instance:

• \(3z^2\) and \(5z^2\) are like terms because they both contain \(z^2\).
• \(-4z\) and \(2z\) are also like terms because they both have the variable \(z\).

Remember, the coefficients (the numerical parts) can be different, but the variables (and their exponents) must match.
When combining like terms:

• Add or subtract their coefficients only.
• The resulting term keeps the same variable and power.

For example, if we have the expression: \(-4z^3 + 3z^3\), both terms are like terms, and combining them gives us \(-z^3\). This step is essential for simplifying polynomial expressions accurately.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They form the foundation of algebra.Each expression is made up of terms, which can include:

• Constants, such as \(7\) or \(-3\).
• Variables, like \(z\).
• Products of constants and variables, such as \(2z^2\).

Polynomials are a type of algebraic expression where variables have non-negative integer exponents.
Working with these expressions involves basic operations like addition, subtraction, multiplication, and sometimes division. The ability to correctly manipulate these expressions is crucial in simplifying mathematical problems and solving equations.
In problems like the one provided, understanding these components and operations allows us to seamlessly perform calculations like polynomial addition and subtraction.