Pregunta
Compare and contrast the following polynomials:
Factor
.
Make a conjecture about how you could apply the method you used to factor
to factor
.
Factor
Make a conjecture about how you could apply the method you used 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.
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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
.
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