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Chapter 5: Problem 48

Arrange in ascending powers of \(x .\) $$ 7-x+3 x^{6}+2 x^{4} $$

Short Answer

Expert verified
7 - x + 2x^4 + 3x^6

Step by step solution

01

Identify all terms with their powers

List each term along with its corresponding power of \(x\): 1. \(7\) (power of \(x\) is 0)2. \(-x\) (power of \(x\) is 1)3. \(2x^4\) (power of \(x\) is 4)4. \(3x^6\) (power of \(x\) is 6)
02

Arrange terms in ascending order of powers

Rearrange the terms from lowest to highest power of \(x\):\[ 7 - x + 2x^4 + 3x^6 \]

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Key Concepts

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

Polynomials
Polynomials are algebraic expressions that consist of variables and coefficients, connected through addition, subtraction, and multiplication. They do not include division by variables or exponential functions. A polynomial can have one or more terms.
Some key points to understand about polynomials:
  • The degree of a polynomial is determined by the highest power of the variable in the expression.
  • Each term in a polynomial consists of a coefficient and a variable raised to a power.
  • Standard form of a polynomial arranges the terms in descending order of their powers.
Polynomials are essential because they appear across various fields in mathematics and science, providing a way to model and solve problems involving curved lines and surfaces.
Sorting Polynomials
Sorting polynomials usually means arranging the terms of the polynomial either in ascending or descending order of their powers. In this exercise, the goal is to arrange the terms in ascending powers of x:
  • Ascending order goes from the lowest power to the highest power.
  • Descending order goes from the highest power to the lowest power.
Here's the detailed breakdown for our exercise:
The original polynomial is:\[ 7 - x + 3x^6 + 2x^4 \].
To arrange this in ascending powers of \(x\), follow these steps:
1. Identify the power of x for each term:
鈥� \(7\) is \(7x^0\)
鈥(-x\) is \(-x^1\)
鈥(2虫镑4袄)
鈥(3虫镑6袄)
2. Sort the terms by the power of \(x\): \[7, -x, 2x^4, 3x^6\].
3. Rewrite the polynomial: \[7 - x + 2x^4 + 3x^6\].
Algebraic Expressions
An algebraic expression is a mathematical phrase that uses numbers, variables (letters), and operation symbols. Unlike equations, algebraic expressions do not have equal signs.

Key points about algebraic expressions:
  • They can be as simple as a single number or variable, or a combination of many.
  • Operations used include addition, subtraction, multiplication, and division.
  • They form the building blocks for equations and functions.
In this exercise, we worked with an algebraic expression in the form of a polynomial. We identified the powers of \(x\) in each term and rearranged them in an orderly fashion.

Understanding algebraic expressions is crucial because they simplify complex problems, making it easier to perform operations and solve for unknowns. They also help in modeling real-life situations and in many other areas of mathematics.

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Most popular questions from this chapter

Solve. Three consecutive even integers are such that the square of the first plus the square of the third is \(136 .\) Find the three integers.

Explain how one could write a quadratic equation that has \(-3\) and 5 as solutions.

Mary Louise is attempting to solve \(x^{3}+20 x^{2}+4 x+80=0\) with a graphing calculator. Unfortunately, when she graphs \(y_{1}=x^{3}+20 x^{2}+4 x+80\) in a standard \([-10,10,-10,10]\) window, she sees no graph at all, let alone any \(x\) -intercept. Can this problem be solved graphically? If so, how? If not, why not?

Let \(f(x)=3 x+1\) and \(g(x)=x^{2}-2 .\) Find the following. The domain of \(f\)

Solve. The foot of an extension ladder is 10 ft from a wall. The ladder is 2 ft longer than the height that it reaches on the wall. How far up the wall does the ladder reach?
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