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Chapter 6: Problem 11

Solve each equation. State the number and type of roots. \(3 x+8=0\)

Short Answer

Expert verified
One real and rational root: \(x = \frac{-8}{3}\).

Step by step solution

01

Isolate the variable

The given equation is \(3x + 8 = 0\). To find the value of \(x\), we need to eliminate the constant term (\(+8\)) on the left side. This can be achieved by subtracting 8 from both sides of the equation. This gives us \(3x = -8\).
02

Solve for x

Now, we need to isolate \(x\) to find its value. Since \(3x = -8\), we divide both sides of the equation by 3 to solve for \(x\). This gives us \(x = \frac{-8}{3}\).
03

State the number and type of roots

Since \(x = \frac{-8}{3}\), the equation has one root. It is a rational number as it can be expressed as a fraction. Therefore, the equation has one real and rational root.

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Key Concepts

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

Solving Equations
The process of solving equations involves finding the values of variables that satisfy the given equation. Let's break down the steps:
  • First, identify and move all constant terms to the other side of the equation. We do this by applying the inverse operation - for addition, we use subtraction, and vice versa. This helps in isolating the variable.
  • Next, once your variable terms are isolated, solve for the variable by performing inverse operations. If the variable is multiplied by a number, divide both sides by that number to get the variable by itself.
Remember to perform each operation on both sides of the equation to maintain balance. This process applies not only to linear equations but to equations in general.
Real Numbers
Real numbers are an essential concept in mathematics encompassing all numbers that can be found on the number line. They include:
  • Natural numbers: like 1, 2, 3...
  • Whole numbers: similar to natural numbers but including zero.
  • Integers: which cover positive and negative numbers, including zero.
  • Rational numbers: numbers like \(\frac{1}{2}\) or \(\frac{-8}{3}\) that can be expressed as fractions.
  • Irrational numbers: numbers that cannot be precisely expressed as fractions, such as \(\pi \) and \(\sqrt{2}\).
In solving equations, real numbers describe the set of possible solutions or roots. The solution \(x = \frac{-8}{3}\) falls into the category of rational numbers, a subset of real numbers. Understanding this concept helps identify the type of solutions you may encounter.
Rational Root Theorem
The Rational Root Theorem is a useful tool in algebra for determining possible rational roots of a polynomial equation. This theorem implies:
  • If a polynomial equation with integer coefficients has a rational solution or root, then the rational root can be expressed as \(\frac{p}{q}\), where:
  • \(p\) is a factor of the constant term.
  • \(q\) is a factor of the leading coefficient.
In the context of our linear equation \(3x + 8 = 0\), although solving for \(x\) doesn't directly apply the Rational Root Theorem, we can understand that the solution \(x = \frac{-8}{3}\) aligns with what we'd expect from a rational solution. Knowing this theorem can help when dealing with more complex polynomial equations.

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Most popular questions from this chapter

Graph each polynomial function. Estimate the \(x\) -coordinates at which the relative maxima and relative minima occur. State the domain and range for each function. $$ f(x)=x^{4}-8 x^{2}+10 $$

Factor completely. If the polynomial is not factorable, write prime. $$ c^{3}-216 $$

For Exercises \(11-18,\) complete each of the following. a. Graph each function by making a table of values. b. Determine the consecutive integer values of \(x\) between which each real zero is located. C. Estimate the \(x\) -coordinates at which the relative maxima and relative minima occur. $$ f(x)=-x^{3}-4 x^{2} $$

If \(p(x)=2 x^{2}-5 x+4\) and \(r(x)=3 x^{3}-x^{2}-2,\) find each value. $$ 2\left[p\left(x^{2}+1\right)\right]-3 r(x-1) $$

For Exerises \(26-31\) , complete each of the following. a. Graph each funnction by making a table of values. b. Determine the consecutive integer values of \(x\) between which each real zero is located. C. Estimate the \(x\) -coordinates at which the relative maxima and relative minima occur. $$ f(x)=-x^{4}+5 x^{2}-2 x-1 $$
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