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Chapter 5: Problem 46

Combine like terms. Write all answers in descending order. \(3 x^{4}-7 x+x^{4}-2 x^{2}\)

Short Answer

Expert verified
The expression in descending order is \(4x^4 - 2x^2 - 7x\).

Step by step solution

01

- Identify like terms

Examine the given expression and identify the like terms. Like terms have the same variable raised to the same power. The expression given is \(3x^{4} - 7x + x^{4} - 2x^{2}\).The like terms are as follows: - Terms with \(x^4\): \(3x^{4}\) and \(x^{4}\)- Terms with \(x\): \(-7x\)- Terms with \(x^2\): \(-2x^{2}\)
02

- Combine the like terms

Combine the like terms by adding or subtracting their coefficients.- For \(x^4\): \( 3x^{4} + x^{4} = 4x^{4} \)- For \(x^2\): There is only one term: \(-2x^{2}\)- For \(x\): There is only one term: \(-7x\)
03

- Write the expression in descending order

Arrange the terms in descending order of their exponents. The combined expression becomes:\(4x^{4} - 2x^{2} - 7x\)

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Key Concepts

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

Polynomials
A polynomial is a mathematical expression made up of variables and constants, combined using addition, subtraction, multiplication, and non-negative integer exponents. Polynomials can have multiple terms.

For instance, in the expression we dealt with: \(3x^{4} - 7x + x^{4} - 2x^{2}\), each part separated by the plus or minus sign is called a term. So, there are four terms in this polynomial.

Every polynomial has a degree, which is the highest power of the variable in the polynomial. In our case, the highest power of \(x\) is 4, so the degree of the polynomial is 4.

Polynomials are fundamental in algebra, and understanding them is key to learning more advanced math concepts.
Like Terms
Like terms are terms that have identical variable parts, including the same exponents. Only the coefficients (the numerical part) can be different.

In our example: \(3x^{4} - 7x + x^{4} - 2x^{2}\), you can see that:
  • The terms \(3x^{4}\) and \(x^{4}\) are like terms because they both contain \(x^{4}\).
  • The term \(-7x\) stands alone but could still combine with another term of \(x\) if it were present.
  • The term \(-2x^{2}\) has no other like term in this expression, but if there were another \(x^{2}\) term, they could be combined.

Combining like terms simplifies polynomials, making them easier to work with.
Descending Order
To write a polynomial in descending order, you need to arrange its terms from the highest power to the lowest power of the variable.

From our combined polynomial: \(4x^{4} - 2x^{2} - 7x\), we list the terms starting with the term with the highest exponent:
  • First, the term with \(x^{4}\), which is \(4x^{4}\).
  • Next, the term with \(x^{2}\), which is \(-2x^{2}\).
  • Finally, the term with \(x\), which is \(-7x\).

Writing in descending order is standard practice as it provides a neat and organized way to represent polynomials, making them clearer to read and easier to manage in further calculations or equations.

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Most popular questions from this chapter

Evaluate. $$ -x^{2}-5 x+3, \text { for } x=-2 $$

To prepare for Section \(5.2,\) review operations with integers (Sections \(1.5-1.7)\) Perform the indicated operations. $$ -3(5) $$

Review factoring expressions and solving equations. $$ 4 x=0 $$

Simplify. Assume that no denominator is zero and that \(0^{0}\) is not considered. $$ \left(\frac{x^{3}}{y^{2} z}\right)^{5} $$

Find a value of the variable that shows that the two expressions are not equivalent. Answers may vary. $$ 3 x^{2} ;(3 x)^{2} $$
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