Question

In Exercises 25 and 26, describe and correct the error in performing the operation. \(\left(x^2-3 x+4\right)-\left(x^3+7 x-2\right)\) \(=x^2-3 x+4-x^3+7 x-2\) \(=-x^3+x^2+4 x+2\)

Short answer

The error lies in the subtraction of the \(7x\) term from the second polynomial. When corrected, the final answer is \(-x^3+x^2-4x+6\).

Step-by-step solution

Spot the error

When the second polynomial is subtracted from the first one, every term in the second polynomial should be multiplied by \(-1\). This is correctly done for the \(x^3\) term and for the \(-2\) term, but it seems to be incorrectly done for the \(+7x\) term. This term should become \(-7x\) after the subtraction, but in the solution presented, it's added, becoming \(+7x\).

Perform the correct subtraction

The correct approach would be to subtract every term in the second polynomial from the first one, switchng their signs. The correct expression after subtraction would be \(x^2-3x+4-x^3-7x+2\) which simplifies to \(-x^3+x^2-4x+6\) instead of \(-x^3+x^2+4 x+2\).

Simplify the expression

Once we get the correct polynomial expression after subtraction, it needs to be simplified. Collect like terms: \(-x^3+x^2-4x+6\). This is the correct, simplified expression.

Key concepts

Subtracting Polynomials

When dealing with subtracting polynomials, it's crucial to understand that subtraction affects every term of the polynomial that is being subtracted. Picture each term as having its own set of parentheses around it. Once you decide to subtract, you flip the sign of each term in the second polynomial before you combine it with the first.

For example, if we have two polynomials, such as \( A(x) = \color{blue}{x^2 - 3x + 4} \) and \( B(x) = \color{red}{x^3 + 7x - 2} \) and we want to subtract B from A (written \(A(x) - B(x)\)), we need to reverse the sign of every term in B. So \(\color{red}{x^3 + 7x - 2}\) becomes \(\color{red}{-x^3 - 7x + 2}\). Now, simply combine like terms with those from A. An error often occurs if we neglect to change the sign of one or more terms, which can alter the outcome significantly.

Let's apply this to our exercise: \(\color{blue}{x^2 - 3x + 4}\) minus \(\color{red}{x^3 + 7x - 2}\) incorrectly became \(x^2 - 3x + 4 - x^3 + 7x - 2\), but it should be \(x^2 - 3x + 4 - x^3 \mathbf{- 7x} + 2\) after correctly flipping the signs. This is a key step in polynomial subtraction to avoid mistakes.

Polynomial Operations

The operations on polynomials, such as addition, subtraction, multiplication, and division, are foundational to algebra. While these operations may appear similar to those you'd perform on regular numbers, it's the 'like terms' concept in polynomials that make these operations unique. Every term in a polynomial is made of variables raised to a power and their coefficients.

During polynomial subtraction, as with our example, align terms with the same exponents and change the signs of the subtracted polynomial before combining them. When adding polynomials, simply combine like terms without changing signs. For multiplication, you would apply the distributive property, multiplying each term across the polynomials, also watching for like terms to combine. Division, often the most complex, can involve long division or synthetic division if applicable.

Understanding these operations is beneficial not only for single exercises but for grasping more advanced topics in algebra, calculus, and beyond.

Algebraic Expressions

At the heart of algebra are algebraic expressions. These are mathematical phrases that include numbers, variables, and operations. A polynomial is a type of algebraic expression with terms that are variables raised to whole number exponents, or constants. For example, \(3x^2 - 6x + 9\) is an algebraic expression where \(3x^2\) is a term, \(x\) is a variable, and numbers like 3 and 9 are constants.

Working with these expressions involves simplifying them using the properties of arithmetic and understanding how to properly manipulate the terms. When simplifying an expression, like when we subtract polynomials, it's essential to perform operations systematically — ensuring terms are combined appropriately and the integrity of the expression is maintained. Algebraic expressions are the building blocks for equations and functions, so mastering their manipulation opens the door to solving more complex mathematical problems.