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Chapter 11: Problem 56

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

Short Answer

Expert verified
The third-degree equation is x^3 - 3x^2 - 4x + 12.

Step by step solution

01

- Understand the Roots

The given solutions to the equation are -2, 2, and 3.
02

- Write the Factors

Each root corresponds to a factor of the equation. Therefore, the factors are: (x + 2), (x - 2), and (x - 3).
03

- Set Up the Equation

Multiply the factors to get the polynomial equation: P(x) = (x + 2)(x - 2)(x - 3).
04

- Expand the Product

First, multiply the first two factors: (x + 2)(x - 2) = x^2 - 4. Next, multiply the result by the third factor: (x^2 - 4)(x - 3).
05

- Perform the Multiplication

Distribute x^2 - 4 through (x - 3): x^2(x - 3) - 4(x - 3) = x^3 - 3x^2 - 4x + 12. Thus, the polynomial equation is 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
Polynomial roots are the values of the variable that make the polynomial equal to zero. For a third-degree polynomial, there will be up to three roots. In the given exercise, we have three specific roots: -2, 2, and 3. When these values are plugged into the polynomial equation, the result should be zero. Understanding the roots is fundamental because it helps us in constructing the polynomial.
Factoring Polynomials
Factoring polynomials involves breaking down a polynomial into simpler 'factors' that, when multiplied together, give back the original polynomial. Given the roots -2, 2, and 3: each root can be associated with a factor. For example:
  • -2 turns into the factor (x + 2)
  • 2 turns into the factor (x - 2)
  • 3 turns into the factor (x - 3)
By multiplying these factors together: (x + 2)(x - 2)(x - 3), we get a polynomial which has these roots.
Expanding Products
Expanding products involves multiplying factors together to form a polynomial. In our example, we start with the factors (x + 2), (x - 2), and (x - 3). We first multiply the first two factors:
  • (x + 2)(x - 2) = x^2 - 4
This result, x^2 - 4, is then multiplied by the third factor:
  • (x^2 - 4)(x - 3)
To achieve the final polynomial, which is a third-degree equation.
Distributive Property
The distributive property is used to simplify algebraic expressions and it states that a(b + c) = ab + ac. In our exercise, we use the distributive property to multiply (x^2 - 4) by (x - 3). Here's how it works step-by-step:
  • First, multiply x^2 by (x - 3): x^2(x - 3) = x^3 - 3x^2
  • Next, multiply -4 by (x - 3): -4(x - 3) = -4x + 12
  • Finally, sum all these products to get the expanded form: x^3 - 3x^2 - 4x + 12
Applying the distributive property correctly ensures that we do not miss any terms in our polynomial expansion.

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Most popular questions from this chapter

Let \(f(x)=(x-5)^{2} .\) Find \(x\) such that \(f(x)=16\).

Let \(f(t)=(t+6)^{2} .\) Find \(t\) such that \(f(t)=15\).

Write an equation for a function having a graph with the same shape as the graph of \(f(x)=\frac{3}{5} x^{2},\) but with the given point as the vertex. $$ (-4,-2) $$

Solve. $$ t^{2}=144 $$

Solve. $$ 25 y^{2}+16=0 $$
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