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Chapter 5: Problem 46

Subtract. $$ \left(8 x^{5}+3 x^{4}+x-1\right)-\left(8 x^{5}+3 x^{4}-1\right) $$

Short Answer

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x

Step by step solution

01

Write the expression

Start by writing the given expression: \((8x^5 + 3x^4 + x - 1) - (8x^5 + 3x^4 - 1)\).
02

Distribute the negative sign

Distribute the negative sign to each term in the second polynomial: \(8x^5 + 3x^4 + x - 1 - 8x^5 - 3x^4 + 1\).
03

Combine like terms

Now combine the like terms in the expression: \((8x^5 - 8x^5) + (3x^4 - 3x^4) + x + (-1 + 1)\).
04

Simplify

Simplify each group of like terms: \(0 + 0 + x + 0 = x\). The simplified answer is \(x\).

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Key Concepts

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

Distributing Negative Sign
When subtracting polynomials, it's crucial to distribute the negative sign to each term inside the parentheses of the second polynomial. This means you change the sign of every term within the parentheses. For instance, if you have (8x^5 + 3x^4 - 1) inside the parentheses, distributing the negative sign gives you -8x^5, -3x^4, and +1. This step can prevent errors in your calculations and ensure you correctly subtract each term. It's like multiplying each term by -1. In the given exercise, after distributing the negative sign, the expression becomes: 8x^5 + 3x^4 + x - 1 - 8x^5 - 3x^4 + 1.
Like Terms
Combining like terms is a fundamental step in polynomial subtraction. Like terms are terms that have the same variables raised to the same power. For example, in the polynomials 8x^5 and -8x^5, both terms are like terms because they both have the variable x raised to the power of 5. When you combine them, you only add or subtract their coefficients. Identifying and combining like terms helps to simplify your polynomial expression. After distributing the negative sign, the next step is to combine the terms with the same powers of x: (8x^5 - 8x^5), (3x^4 - 3x^4), and the constants (-1 + 1). The other term x does not have a like term to combine with in this case.
Simplification
Simplification involves reducing the polynomial expression to its simplest form. After combining like terms, you'll often find that some terms cancel each other out. For example, in (8x^5 - 8x^5), the terms result in 0 because 8 minus 8 equals 0. Similarly, (3x^4 - 3x^4) equals 0 and (-1 + 1) equals 0. This leaves you with the term x, which does not change because there is no like term to combine with. After simplifying all the terms in the given exercise, the expression reduces to x. So, the answer is x, which is the simplest form of the polynomial after subtraction.

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Most popular questions from this chapter

F(x)\( and \)g(x)\( are as given. Find a simplified expression for \)F(x)\( if \)F(x)=(f / g)(x)$. $$ f(x)=8 x^{3}+27, g(x)=2 x+3 $$

Yearly Depreciation. An investment \(P\) that drops in value at the yearly rate \(r\) drops in value to $$ P(1-r)^{t} $$ after \(t\) years. Find a polynomial that can be used to determine the value to which \(P\) has dropped after 2 years.

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