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Chapter 8: Problem 56

Write a third-degree equation having the given numbers as solutions. $$-2,2,3$$

Short Answer

Expert verified
The third-degree polynomial is \( x^3 - 3x^2 - 4x + 12 \).

Step by step solution

01

- Understand the Root Form

Given roots of the equation are -2, 2, and 3. For an equation with roots, we use the form: \[ (x - r_1)(x - r_2)(x - r_3) = 0 \] Here, \[ r_1 = -2, \quad r_2 = 2, \quad r_3 = 3 \]
02

- Substitute the Roots into the Polynomial Form

Substitute the given roots into the polynomial form: \[ (x + 2)(x - 2)(x - 3) = 0 \]
03

- Expand the Binomials

First, expand \( (x + 2)(x - 2) \) using the difference of squares formula: \[ (x + 2)(x - 2) = x^2 - 4 \]Next, multiply the result by \( (x - 3) \): \[ (x^2 - 4)(x - 3) \]
04

- Expand the Remaining Expression

Expand the expression by multiplying each term in \( x^2 - 4 \) by each term in \( (x - 3) \): \[ (x^2 - 4)(x - 3) = x^3 - 3x^2 - 4x + 12 \]

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Key Concepts

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

Polynomial Roots
When you solve a polynomial equation, you are looking for its roots. Roots are values of the variable that satisfy the equation, making the entire expression equal zero.
For a polynomial of the form \(x^3 - 3x^2 - 4x + 12 = 0\), the roots are the values of \(x\) that make the equation true. This includes both real and complex numbers.
In your given problem, the roots are \(-2\), \(2\), and \(3\).
These roots are substituted back into the factored form of the polynomial to form the equation.
Difference of Squares
The difference of squares is a special algebraic identity. It states that \(a^2 - b^2 = (a + b)(a - b)\).
This is crucial for simplifying expressions and solving polynomial equations.
In the given exercise, we use this property to simplify \((x + 2)(x - 2)\).
By applying the difference of squares identity, we can rewrite \((x + 2)(x - 2)\) as \(x^2 - 4\).
Simplifying polynomials using identities like the difference of squares makes the process easier and reduces the risk of error.
Expanding Binomials
Expanding binomials means multiplying them out to remove parentheses and simplify the expression.
For instance, consider \((x^2 - 4)(x - 3)\).
To expand this, you distribute each term in \(x^2 - 4\) by each term in \(x - 3\).
This gives: \(x^2 \times x = x^3\), \(-3 \times x^2 = -3x^2\), \(-4 \times x = -4x\), and \(-4 \times -3 = 12\).
Adding these results together, we get the expanded form: \(x^3 - 3x^2 - 4x + 12\).
Expanding is an essential skill in algebra to simplify complex polynomials.
Algebraic Equations
Algebraic equations involve variables and constants connected by arithmetic operations.
They can be linear, quadratic, cubic, or of higher degrees, depending on the highest power of the variable.
In this exercise, we are dealing with a third-degree (cubic) equation because the highest power of the variable \(x\) is \(3\).
Cubic equations can have up to three real roots.
Solving algebraic equations usually involves factoring, using special algebraic identities, or applying methods like the quadratic formula for second-degree equations.
Understanding the structure and properties of algebraic equations helps to tackle various mathematical problems effectively.

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Most popular questions from this chapter

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