Problem 46 - Beginning and Intermediate Algebra

Question

Subtract. $$\left(16 w^{3}+9 w-7\right)-\left(27 w^{3}-3 w-4\right)$$

Step-by-Step Solution

Answer

The subtraction of the given polynomials is: \( -11w^3 + 12w - 3 \).

Step 1: Arrange polynomials with like terms

Write both polynomials with their like terms aligned: \[ \begin{array}{@{}c@{\;}c@{\;}c@{\;}c@{\;}c} & 16w^3 & + 9w & - 7 & \\ - & 27w^3 & - 3w & - 4 & \\ \cline{1-5} \end{array} \]

Step 2: Subtract the coefficients of the like terms

Subtract the coefficients of the like terms in each column: \[ \begin{array}{@{}c@{\;}c@{\;}c@{\;}c@{\;}c} & 16w^3 & + 9w & - 7 & \\ - & 27w^3 & - 3w & - 4 & \\ \cline{1-5} & (16-27)w^3 & + (9-(-3))w & - (7-4) & \end{array} \]

Step 3: Simplify the terms

Simplify the coefficients in each term: \[ \begin{array}{c@{\;}c@{\;}c} (16-27)w^3 & + (9-(-3))w & - (7-4) \\ (-11)w^3 & + 12w & - 3 \end{array} \]

Step 4: Write the final answer

Combine the simplified terms to get the final answer: \[ -11w^3 + 12w - 3 \] So, the subtraction of the given polynomials is: \( -11w^3 + 12w - 3 \).

Key Concepts

Polynomial Operations

Polynomial operations include all the arithmetic processes you can perform on polynomials, such as addition, subtraction, multiplication, and division. These operations are similar to those of arithmetics, but you're dealing with terms consisting of variables raised to powers.

• Addition: Combine the like terms by adding their coefficients.
• Subtraction: Similar to addition, but involves subtracting the coefficients of like terms.
• Multiplication: Distribute each term of the first polynomial to every term of the second one, then simplify.
• Division: Similar to long division in numbers but involves considering variable terms.

Understanding these operations is key to manipulating algebraic expressions and solving algebraic equations effectively. In essence, polynomial operations allow you to manage complex expressions in manageable ways. This makes tackling algebra problems much more straightforward.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables (like w, x, y), and arithmetic operators (+, -, *, /). They don't have an equals sign, differentiating them from equations. In this exercise, your expression looks like polynomial expressions, which feature powers on the variables.

• Terms: Building blocks of expressions that may include constants, variables, and coefficients (e.g., 16\(w^3\)).
• Like Terms: Terms whose variables are the same and that can be combined (e.g., 16\(w^3\) and 27\(w^3\)).

Mastering algebraic expressions requires understanding these concepts so you can simplify and solve them. They form the foundation of algebra and are encountered frequently within broader mathematical contexts.

Coefficients

Coefficients are the numerical part of the terms in algebraic expressions that precede the variables. They are super important because they indicate how many times to multiply the variable part of a term. So, in 16\(w^3\), 16 is the coefficient.

• It shows how much of the variable, in this case, \(w^3\), you have.
• In subtraction, like in our solution, coefficients determine the outcomes when like terms are combined.

For instance, subtracting the coefficients for \(w^3\) in our problem involved doing 16 - 27. The outcome, -11, becomes the new coefficient for the resulting term, \(-11w^3\). Fully grasping how coefficients work allows you to manipulate expressions efficiently.

Subtraction of Like Terms

Subtraction of like terms is a crucial skill when working with polynomials. It involves aligning terms with the same variables and powers, then subtracting their coefficients. In this exercise:

• We aligned \(16w^3\) with \(27w^3\) and subtracted: 16 - 27 to get \(-11w^3\).
• Aligned \(9w\) with \(-3w\) and subtracted: 9 - (-3) to get \(12w\).
• Aligned the constant terms and subtracted: -7 - (-4) to get -3.

This simplification provides a cleaner expression that is more manageable. Recognizing and manipulating like terms is at the heart of polynomial subtraction, enabling you to simplify even very complex algebraic expressions with ease.