Problem 55 Add or subtract. $$\left(4 t^{... [FREE SOLUTION]

Chapter 11: Problem 55

Add or subtract. $$\left(4 t^{2}+5 t+2\right)-\left(t^{2}-3 t-8\right)$$

Short Answer

Expert verified

The solution to the subtraction of these two trinomials is $3t^2 + 8t + 10$.

Step by step solution

Rewrite the Equation

It's necessary to rewrite the equation, removing the parentheses. Remember that a minus sign before a parenthesis means to change the sign of each term inside the parenthesis. The equation then looks like this: $4t^2+5t+2 - t^2 + 3t + 8$

Combine Like Terms

Then proceed by adding together the like terms. The like terms are those which have the same variable raised to the same power. Here, these are $4t^2 - t^2$, $5t + 3t$ and $2 + 8$. The addition of these terms results in the combined expression: $3t^2 + 8t + 10$

Write down the final result

The final result of this polynomial subtraction is: $3t^2 + 8t + 10$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combining Like Terms

To solve polynomial expressions, an important skill is combining like terms. The process involves grouping terms in an expression that share the same variables raised to the same power. Essentially, you're looking to simplify equations by adding or subtracting these similar components.

First, identify terms with the same variables and exponents. For instance, in the expression $4t^2$ and $t^2$, both terms include $t^2$, which makes them like terms.
Next, perform the arithmetic operations. For our example, you subtract the coefficients $4$ and $1$ from the $t^2$ terms, resulting in $3t^2$.

Combining like terms is a fundamental part of this process, ensuring that the polynomial is as simplified as possible. By following these steps, your expressions become more manageable and clearer to interpret.

Polynomial Operations

Polynomial operations include addition, subtraction, multiplication, and division of polynomial expressions. When you subtract polynomials, like in the exercise, there's an important rule: changing the sign of the second polynomial's terms before combining them with the first.
Let's explore this further:

Addition and subtraction of polynomials: Involve combining like terms, as seen earlier. It's crucial for maintaining equation accuracy.
Negation in subtraction: A minus sign before a polynomial impacts all the following terms. Signs change from positive to negative or vice versa. For instance, the negative $t^2$ becomes $-t^2$, and negative $-3t$ turns into positive $+3t$.

This switch in signs is critical as it sets the stage for correctly solving the polynomial subtraction, leading you to the simple and clean finished expression.

Algebraic Expressions

An algebraic expression is made from variables, numbers, and operations such as addition and subtraction. These expressions often need simplifying, especially when they're polynomials like the one in the exercise. Understanding how they work is key to manipulating and simplifying them efficiently.
Consider the characteristics of algebraic expressions:

Variables: These represent unknowns and can vary. In our example, $t$ is the variable.
Coefficients: Numbers that multiple the variables. For instance, in $4t^2$, the number $4$ is the coefficient.
Constants: Number-only terms without variables, like $2$ and $8$.

Understanding these elements helps you deconstruct and reconstruct expressions when solving polynomial equations. You'll find it easier to arrange, modify and simplify complex tasks into manageable segments.

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91影视

Add or subtract. $$\left(4 t^{2}+5 t+2\right)-\left(t^{2}-3 t-8\right)$$

Short Answer

Step by step solution

Rewrite the Equation

Combine Like Terms

Write down the final result

Key Concepts

Combining Like Terms

Polynomial Operations

Algebraic Expressions

One App. One Place for Learning.

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