Question

Use a vertical format or a horizontal format to add or subtract. $$\left(6 t^{2}-19 t\right)-\left(3-2 t^{2}\right)-\left(8 t^{2}-5\right)$$

Short answer

The simplified result of the given expression is \( t^2 - 19t + 2\).

Step-by-step solution

Identify like terms and distribute negatives

First, identify the like terms as they will be added or subtracted: \(6t^2\), \( -2t^2\), and \( -8t^2\) ; \( -19t\), and \( -5\), \(3\) correspondingly.

Distribute Negatives

Next, distribute the negative signs to all terms in the parentheses. The original expression is equivalent to \(6t^2 - 19t - 3 + 2t^2 - 8t^2 + 5\).

Combine like terms

Now combine all like terms. This leads to: \( t^2 - 19t + 2\).

Key concepts

Polynomial Arithmetic

Polynomial arithmetic is the foundation for working with algebraic expressions that include variables raised to different powers. In essence, a polynomial is a mathematical expression that consists of variables (such as t) and coefficients, which are combined using addition, subtraction, multiplication, and non-negative integer exponents.

One of the essential operations in polynomial arithmetic is combining like terms. Like terms are those terms within an expression that have the same variables raised to the same powers. For example, in the expression \(6t^2 + 3t - 4t^2 + 2t\), \(6t^2\) and \( -4t^2\) are like terms, as are \(3t\) and \(2t\).

To simplify such expressions, you add or subtract the coefficients of like terms, which essentially involves combining their 'numerical parts' while keeping the variable parts unchanged. In our exercise, \(6t^2\), \( -2t^2\), and \( -8t^2\) are combined to get the final coefficient for \(t^2\), and similarly for other like terms.

Subtracting Polynomials

Subtracting polynomials is a process that involves taking away the terms of one polynomial from another. To keep track and ensure accuracy, one often arranges the terms vertically by like terms or writes them out horizontally, being careful to align like terms.

The key is to make sure that you're only subtracting coefficients of like terms. When subtracting an entire polynomial, like in the expression \( (6t^2-19t) - (3-2t^2) - (8t^2-5) \), it's important to distribute the subtraction sign across all terms of the polynomials being subtracted. This sometimes means changing the sign of each term in the polynomial after the subtraction sign.

After distributing the subtraction signs, the terms can be grouped by like terms, and their coefficients can be subtracted accordingly. The student-friendly approach would involve writing each step clearly, and possibly using different colors or markings for different sets of like terms, to make the process visual and straightforward.

Distributive Property

The distributive property is an algebraic rule that is essential for working with expressions that involve parentheses. It allows you to multiply a single term by each term inside a parenthesis. Conversely, when a negative sign precedes a parenthesis, you effectively distribute the negative sign, or multiply by -1, to each term inside that parenthesis.

For example, given \( - (a + b)\), using the distributive property, this becomes \( -a - b\), because you distribute the negative sign to both \(a\) and \(b\).

In the context of our exercise, when we have negative signs before the parentheses like in \( -(3 - 2t^2)\) and \( -(8t^2 - 5)\), we apply the distributive property by multiplying the negative sign across each term inside the parentheses. This turns \(3 - 2t^2\) into \( -3 + 2t^2\) and \(8t^2 - 5\) into \( -8t^2 + 5\), allowing us to then proceed with combining like terms. It's a pivotal concept that often requires a step-by-step demonstration to help learners fully grasp the changes to the expressions.