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Chapter 5: Problem 3

Without solving each equation, find the sum and product of the roots. \(x^{2}+x+1=0\)

Short Answer

Expert verified
Sum of the roots: -1; Product of the roots: 1.

Step by step solution

01

Review the Quadratic Formula

The quadratic equation is given by \( ax^2 + bx + c = 0 \). For the equation \( x^2 + x + 1 = 0 \), we identify that \( a = 1 \), \( b = 1 \), and \( c = 1 \). Later this will help us utilize Vieta's formulas.
02

Understanding Vieta's Formulas

Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. For a quadratic equation \( ax^2 + bx + c = 0 \), the sum of the roots \( r_1 + r_2 = -\frac{b}{a} \) and the product \( r_1 \cdot r_2 = \frac{c}{a} \).
03

Calculate the Sum of the Roots

Apply Vieta's formula for sum of the roots: \( r_1 + r_2 = -\frac{b}{a} = -\frac{1}{1} = -1 \). So, the sum of the roots is \(-1\).
04

Calculate the Product of the Roots

Apply Vieta's formula for the product of the roots: \( r_1 \cdot r_2 = \frac{c}{a} = \frac{1}{1} = 1 \). Thus, the product of the roots is \(1\).

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Key Concepts

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

Vieta's Formulas
Vieta's formulas provide an incredible shortcut to understanding the relationship between the roots of a polynomial and its coefficients. Instead of solving the quadratic equation directly to find roots, we can use Vieta's formulas to find their sum and product. This is particularly useful because it saves time and effort when only the sum and product are needed. These formulas are derived from the quadratic equation in the form of \( ax^2 + bx + c = 0 \). Here:
  • \( r_1 + r_2 = -\frac{b}{a} \): This provides the sum of roots.
  • \( r_1 \cdot r_2 = \frac{c}{a} \): This provides the product of roots.

Understanding and applying these simple relationships can greatly enhance your calculations in algebra without having to dive into more complex procedures like the quadratic formula.
Sum of Roots
The sum of the roots of a quadratic equation can be found using Vieta's formula which states \( r_1 + r_2 = -\frac{b}{a} \). This is derived from reorganizing terms in the quadratic equation form \( ax^2 + bx + c = 0 \).In the specific example of the equation \( x^2 + x + 1 = 0 \):
  • Where \( a = 1 \), \( b = 1 \), and \( c = 1 \).
  • Therefore, the sum of the roots is \( r_1 + r_2 = -\frac{1}{1} = -1 \).

This means for any quadratic in the form \( ax^2 + bx + c \), you can quickly compute the sum of its roots without having to find the roots explicitly. It's a simple yet powerful tool.
Product of Roots
The product of the roots of a quadratic equation also connects to its coefficients using Vieta's formulas. Specifically, the product is given by \( r_1 \cdot r_2 = \frac{c}{a} \).Applying this to the equation \( x^2 + x + 1 = 0 \):
  • With \( a = 1 \), \( b = 1 \), \( c = 1 \).
  • The product of the roots is calculated as \( r_1 \cdot r_2 = \frac{1}{1} = 1 \).

By having this formula at hand, you can easily determine the product of roots for any quadratic equation, simplifying many algebraic problems without directly solving for the roots. This insight can be especially beneficial when dealing with complex polynomials or when rapid calculations are necessary.

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Most popular questions from this chapter

In \(3-18,\) find all roots of each given function by factoring or by using the quadratic formula. $$ f(x)=x^{3}-18 x $$

Write a quadratic equation with integer coefficients for each pair of roots. \(-3,4\)

In \(19-28 :\) a. Find \(\mathrm{f}(a)\) for each given function. b. Is \(a\) a root of the function? $$ \mathrm{f}(x)=5 x^{2}+4 x+1 \text { and } a=-1 $$

In \(19-28 :\) a. Find \(\mathrm{f}(a)\) for each given function. b. Is \(a\) a root of the function? $$ f(x)=x^{3}-2 x+3 \text { and } a=2+i $$

One of the roots is given. Find the other root. \(m^{2}-4 m+n=0 ; 3\)
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