Question

Explain the mistake and work the problem correctly. $$\begin{array}{l} \left(x^{4}-6 x^{2}+3 x\right)-\left(x^{4}+2 x^{3}-4 x\right) \\ \quad=\left(x^{4}-6 x^{2}+3 x\right)+\left(-x^{4}-2 x^{3}+4 x\right) \\ \quad=-2 x^{4}-2 x^{3}-6 x^{2}+7 x \end{array}$$

Short answer

Question: Find the mistake in the given polynomial subtraction expression and then solve the problem correctly: $$(x^{4}-6x^{2}+3x) - (x^{4}+2x^{3}-4x)$$ Answer: The correct result, after fixing the mistake, is $-2x^{3} - 6x^{2} + 7x$.

Step-by-step solution

Write down the given expression

First, let's write down the given expression: $$(x^{4}-6x^{2}+3x) - (x^{4}+2x^{3}-4x)$$

Add the additive inverse of the second polynomial

Now, we need to add the additive inverse of the second polynomial to the first polynomial. Apply the distributive property of multiplication over addition and multiply the second polynomial by -1: $$ (x^{4}-6x^{2}+3x) + (-1)(x^{4}+2x^{3}-4x) $$ Distribute -1 to each term: $$(x^{4}-6x^{2}+3x) + (-x^{4}-2x^{3}+4x)$$

Combine like terms

Now, let's combine like terms by adding the coefficients of each term with the same power of x: $$(x^{4} + (-x^{4})) + (-2x^{3}) + (-6x^{2}) + (3x + 4x)$$ $$ 0x^{4} - 2x^{3} - 6x^{2} + 7x $$ The final result, after correcting the mistake, is: $$ -2x^{3} - 6x^{2} + 7x $$

Key concepts

Combining Like Terms

When dealing with polynomials, an essential skill is combining like terms. This process involves grouping together terms that have the same variable raised to the same power. For example, in an expression like \(3x + 4x\), both terms are "like" because they both contain \(x\) to the power of one.
To combine them, simply add their coefficients (the numbers in front), in this case obtaining \(7x\). Combining like terms simplifies an expression and reduces it to its simplest form.
This process allows us to clearly see the coefficients for each degree of \(x\) and is particularly useful during operations such as addition and subtraction of polynomials.
Keep in mind that only terms with exactly the same variable and exponent can be combined. Terms like \(x^2\) and \(x^3\) do not combine, as they have different exponents, thus remaining separate in expressions.

Distributive Property

The distributive property is a foundational concept in algebra that allows you to expand expressions involving multiplication over addition or subtraction. It’s famously captured by the formula: \(a(b + c) = ab + ac\). This process "distributes" the multiplication of \(a\) across each term in the parenthesis.
In polynomial subtraction, as shown in the example problem, the distributive property is used to effectively change the subtraction of a polynomial into an addition by distributing a negative sign (or -1) across the terms.
This is crucial because it keeps the integrity of each term in the expression intact, while allowing us to apply further simplification techniques like combining like terms seamlessly.

• Helps in expanding and simplifying expressions.
• Ensures that operations are correctly applied to each term.

Understanding the distributive property allows you to tackle complex expressions confidently by breaking them down into manageable pieces.

Additive Inverse

The concept of the additive inverse is fundamental when it comes to understanding subtraction as the addition of opposites. In mathematics, each number has an additive inverse, which when added to the number, results in zero.
For example, the additive inverse of 5 is -5 because \(5 + (-5) = 0\). This property allows subtraction to be rewritten as the addition of the additive inverse.
In the context of polynomials, the additive inverse is used when subtracting one polynomial from another. You "flip the sign" of every term in the polynomial being subtracted, effectively turning a subtraction problem into an addition problem.
This "flipping" is what was done in the original problem to successfully subtract the polynomials:

• Original term \(x^4\) has an additive inverse of \(-x^4\).
• Each term is adjusted, ensuring correct operation without changing the expression's value.

Understanding that subtraction can be turned into addition through additive inverses makes complex polynomial manipulations much simpler.