Pregunta
upstudy study bank question image url

Compare and contrast the following polynomials:


Factor .
Make a conjecture about how you could apply the method you used to factor
to factor .

Ask by Turner Hampton. in the United States
Jan 29,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

Responder

Both polynomials are differences of squares. factors to . Similarly, can be factored by recognizing it as , which factors to . Further factoring gives , so fully factors to .

Solución

Certainly! Let’s delve into the comparison and contrast of the polynomials and , factor , and explore how the factoring method can be extended to .

1. Comparing and Contrasting and

Similarities:
  • Structure: Both polynomials are difference of terms, specifically expressing the subtraction of two terms.
  • Degree: is a second-degree (quadratic) polynomial, while is a fourth-degree polynomial. Both have even degrees.
  • Potential for Factoring: Both can be factored using the difference of squares formula.
Differences:
  • Degree and Complexity:
    • is quadratic and relatively straightforward to factor.
    • is quartic (fourth-degree) and may require multiple factoring steps.
  • Intermediate Terms:
    • involves only two terms.
    • can be viewed as , revealing a nested difference of squares.

2. Factoring

The polynomial is a classic example of a difference of squares, which follows the formula:
Applying this to :
  • Identify and :
    • (since )
  • Factor using the difference of squares:
Factorization Result:

3. Conjecture for Factoring

Observation:
  • can be rewritten as , which is itself a difference of squares.
Conjecture:
  • Step 1: Apply the difference of squares formula to :
  • Step 2: Notice that is again a difference of squares:
  • Step 3: Combine the factors:
Extended Factoring:
  • While is a sum of squares and cannot be factored over the real numbers, it can be factored over the complex numbers as:
Final Factorization (Over Complex Numbers):
Conclusion:
  • By recognizing that is a difference of squares, we can apply the same factoring technique used for recursively. This method effectively breaks down higher-degree polynomials into products of lower-degree polynomials, simplifying the factoring process.

Summary:
  • Both and are differences of squares and can be factored accordingly.
  • The factoring process for serves as a foundational method that can be extended to factor higher-degree polynomials like by identifying and applying the difference of squares multiple times.

Respondido por UpStudy AI y revisado por un tutor profesional

error msg
Explicar
Simplifique esta solución

Bonus Knowledge

The polynomial is a difference of squares, which can be factored into . This method works because it follows the identity , where and . The resulting factors represent the points where the polynomial intersects the x-axis.
Now, when we consider , we can also recognize it as a difference of squares: . By applying the same factoring technique, we first factor it into and then notice that can be further factored as . This showcases how the method used for can indeed be extended to factor more complex polynomials like .

¡Prueba Premium ahora!
¡Prueba Premium y hazle a Thoth AI preguntas de matemáticas ilimitadas ahora!
Quizas mas tarde Hazte Premium
Estudiar puede ser una verdadera lucha
¿Por qué no estudiarlo en UpStudy?
Seleccione su plan a continuación
Prima

Puedes disfrutar

Empieza ahora
  • Explicaciones paso a paso
  • Tutores expertos en vivo 24/7
  • Número ilimitado de preguntas
  • Sin interrupciones
  • Acceso completo a Respuesta y Solución
  • Acceso completo al chat de PDF, al chat de UpStudy y al chat de navegación
Básico

Totalmente gratis pero limitado

  • Solución limitada
Bienvenido a ¡Estudia ahora!
Inicie sesión para continuar con el recorrido de Thoth AI Chat
Continuar con correo electrónico
O continuar con
Al hacer clic en "Iniciar sesión", acepta nuestros términos y condiciones. Términos de Uso & Política de privacidad