Problem 60 - Beginning and Intermediate Algebra

Question

Subtract. $$\begin{array}{r} 7 y^{4}+y^{3}-10 y^{2}+6 y-2 \\ \--2 y^{4}+y^{3} & -4 y+1 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The short answer is: \(9y^4 - 10y^2 + 10y -3\)

Step 1: Write down the given polynomials

We have twopolynomials that we need to subtract: First polynomial: \(7y^4 + y^3 - 10y^2 + 6y - 2\) Second polynomial: \(-2y^4 + y^3 - 4y + 1\)

Step 2: Align the terms with the same exponents

We will now arrange both polynomials vertically, aligning the terms with the same exponents together: $$\begin{array}{ccc} 7y^4&+y^3&-10y^2&+6y&-2\\ -2y^4&+y^3&0&-4y&+1\\ \end{array}$$

Step 3: Perform the subtraction

Now, subtract the second polynomial's terms from the corresponding terms of the first polynomial: $$\begin{array}{ccc} (7y^4-(-2y^4))&+(y^3-y^3)&+(-10y^2-0)&+(6y-(-4y))&+(-2-1)\\ 9y^4 & & -10y^2& +10y & -3 \end{array}$$

Step 4: Write down the result

Combine the resulting terms together to obtain the result of the subtraction: Final result: \(9y^4 - 10y^2 + 10y -3\)

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of numbers, symbols, and operations that represent a certain mathematical concept or quantity. These expressions often use variables, like \( y \) in our example, to stand for numbers we don’t necessarily know. This can look as simple as \( 3x \) or more complex with several terms added or subtracted, like \( 7y^4 + y^3 - 10y^2 + 6y - 2 \). When we say 'combine like terms,' we're referring to combining all terms with identical variables raised to the same power. This makes simplifying algebraic expressions easier. Knowing how to work with algebraic expressions is crucial for tackling more advanced mathematical concepts, including polynomial subtraction.

Polynomial Operations

Polynomial operations include addition, subtraction, multiplication, and division of polynomials. Subtracting polynomials, like in this exercise, involves understanding that polynomials are expressions consisting of variables and coefficients organized in terms where the exponents are whole numbers.
In polynomial subtraction, the key is to subtract each term of one polynomial from the corresponding term of another polynomial. This involves:

• Reversing the sign of each term in the second polynomial.
• Aligning terms that have the same exponent.
• Subtracting the coefficients of these aligned terms.

This process is similar to regular subtraction but applied term by term. So, for instance, subtracting \(-2y^4\) from \(7y^4\) effectively becomes adding \(2y^4\) to \(7y^4\), which results in \(9y^4\). Mastering these operations sets a solid foundation for all sorts of algebraic manipulations.

Polynomial Alignment

Polynomial alignment helps us clearly see which terms need to be combined or subtracted. It's a way to organize each term by their degree (the largest exponent of the variable).
For polynomial subtraction, proper alignment means writing each polynomial vertically, matching terms with the same powers together. It’s like lining up numbers in regular subtraction to ensure each term is properly aligned with others having the same degree:

• Add placeholders for missing degrees, such as adding \( +0 \) when a particular power is not present.
• Aim for neat, column-wise arrangement to avoid confusion.

With good polynomial alignment, performing operations becomes clearer and less prone to error. This results in a more organized subtraction process and produces accurate results, as clearly observed in the final step of this exercise, which led to the correct result of \(9y^4 - 10y^2 + 10y - 3\).