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Chapter 4: Problem 71

Simplify the algebraic expressions for the following problems. $$ 7 x^{4}-15 x^{4} $$

Short Answer

Expert verified
Question: Simplify the algebraic expression: \(7x^4 - 15x^4\). Answer: \(-8x^4\)

Step by step solution

01

(Step 1: Identify Like Terms)

To simplify the algebraic expression, first identify the like terms by looking for expressions with the same variable and exponent. In this case, we have two terms involving the variable x raised to the power of 4: $$ 7x^4\quad\text{and}\quad-15x^4 $$
02

(Step 2: Combine the Like Terms)

Since both terms are like terms, we can combine them by performing the subtraction. To do this, we subtract the coefficients of the like terms: $$ 7x^4 - 15x^4 = (7 - 15) x^4 $$
03

(Step 3: Perform the Subtraction)

Now perform the subtraction of the coefficients: $$ (7 - 15) x^4 = -8x^4 $$
04

(Final Answer)

The simplified algebraic expression is: $$ -8x^4 $$

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Key Concepts

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

Understanding Like Terms
Like terms are the building blocks that make simplifying algebraic expressions possible. To identify like terms, you look for terms with the same variable raised to the same power. For example, the terms \(7x^4\) and \(-15x^4\) are considered like terms because they both contain the variable \(x\) raised to the fourth power.
It's important to note that the coefficients, the numbers in front of the variables, do not have to be the same for the terms to be like terms. As long as the variable and the exponent match, you can combine these terms.
To combine them, you simply add or subtract the coefficients while keeping the variable and its exponent the same.
The Role of Coefficients
Coefficients are the numerical part of terms that appear with variables in an algebraic expression. In the expression \(7x^4 - 15x^4\), the numbers 7 and -15 are the coefficients of the terms \(x^4\).
They play a critical role when simplifying expressions. While the variable and exponent of like terms remain unchanged through the process of simplification, it is the coefficients that are added or subtracted.
In our example, you subtract the coefficient 15 from 7, which gives you \(-8\), resulting in the simplified term \(-8x^4\). Through this operation, coefficients help determine the new numerical value accompanying the variable term.
Steps in Polynomial Simplification
Polynomial simplification involves a few clear steps that revolve around identifying like terms and combining them correctly. It starts with carefully examining the polynomial to find terms that have the same variable raised to the same power.
  • Identify Like Terms: As in the example \(7x^4 - 15x^4\), check which terms can be combined by matching variables and exponents.

  • Combine Coefficients: After selecting like terms, add or subtract their coefficients. Here, you would calculate \(7 - 15\) to get \(-8\).

  • Rewrite the Expression: Form the simplified expression by pairing the new coefficient with the common variable and exponent, resulting in \(-8x^4\) in our problem.

This systematic approach is what polynomial simplification is all about, and mastering it makes handling algebraic expressions a breeze.

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Most popular questions from this chapter

For the following problems, perform the multiplications and combine any like terms. $$ (2 a+b)\left(5 a^{2}+4 a^{2} b-b-4\right) $$

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