The factors of x2 + 4y2 + 4y - 4xy - 2x - 8 are:
(x - 2y - 4) (x - 2y + 2)
Understanding Polynomial Factorization
The problem asks us to find the factors of the given polynomial expression:
\(\text{x}^2 + 4\text{y}^2 + 4\text{y} - 4\text{xy} - 2\text{x} - 8\)
To factor this complex expression, we can try grouping terms and looking for patterns, specifically algebraic identities or structures that resemble quadratic forms.
Step-by-Step Factorization Process
Let's rearrange the terms to group similar components together, especially those that might form a perfect square trinomial or a recognizable pattern.
The terms \(\text{x}^2\), \(4\text{y}^2\), and \(-4\text{xy}\) suggest an identity of the form \((\text{a} - \text{b})^2 = \text{a}^2 - 2\text{ab} + \text{b}^2\). Here, if we let \(\text{a} = \text{x}\) and \(\text{b} = 2\text{y}\), then \((\text{x} - 2\text{y})^2 = \text{x}^2 - 4\text{xy} + 4\text{y}^2\). Let's group these terms:
\((\text{x}^2 - 4\text{xy} + 4\text{y}^2) - 2\text{x} + 4\text{y} - 8\)
Substitute the identity:
\((\text{x} - 2\text{y})^2 - 2\text{x} + 4\text{y} - 8\)
Now, look at the remaining linear terms: \(-2\text{x} + 4\text{y}\). We can factor out \(-2\) from these terms:
\(-2\text{x} + 4\text{y} = -2(\text{x} - 2\text{y})\)
Substitute this back into the expression:
\((\text{x} - 2\text{y})^2 - 2(\text{x} - 2\text{y}) - 8\)
Notice that the expression now looks like a quadratic equation in terms of \((\text{x} - 2\text{y})\). Let's use a substitution to make it clearer. Let \(u = \text{x} - 2\text{y}\).
The expression becomes:
\(u^2 - 2u - 8\)
Now we need to factor this simple quadratic expression. We look for two numbers that multiply to \(-8\) and add up to \(-2\). These numbers are \(-4\) and \(2\).
So, the factored form of \(u^2 - 2u - 8\) is \((u - 4)(u + 2)\).
Finally, substitute back \(u = \text{x} - 2\text{y}\) into the factored expression:
\((\text{x} - 2\text{y} - 4)(\text{x} - 2\text{y} + 2)\)
These are the factors of the original polynomial expression.
Comparing Factors with Options
Let's compare our derived factors \((\text{x} - 2\text{y} - 4)(\text{x} - 2\text{y} + 2)\) with the given options:
- Option 1: \((\text{x} - 2\text{y} - 4) (\text{x} - 2\text{y} + 2)\)
- Option 2: \((\text{x}^2 - 2\text{y} - 4) (\text{x}^2 - 2\text{y} + 2)\)
- Option 3: \((\text{x} + 2\text{y} - 4) (\text{x} + 2\text{y} + 2)\)
- Option 4: \((\text{x}^2 - 2\text{y} - 4) (\text{x}^2 + 2\text{y} + 2)\)
Our factors match Option 1.
Revision Table: Key Factorization Concepts
| Concept | Description | Example |
|---|---|---|
| Grouping Terms | Rearranging terms to identify patterns or common factors. | \(ax + ay + bx + by = a(x+y) + b(x+y) = (a+b)(x+y)\) |
| Perfect Square Trinomial | \(\text{a}^2 \pm 2\text{ab} + \text{b}^2 = (\text{a} \pm \text{b})^2\) | \(\text{x}^2 - 6\text{x} + 9 = (\text{x} - 3)^2\) |
| Difference of Squares | \(\text{a}^2 - \text{b}^2 = (\text{a} - \text{b})(\text{a} + \text{b})\) | \(4\text{x}^2 - 25 = (2\text{x} - 5)(2\text{x} + 5)\) |
| Factoring Quadratics | Factoring \(\text{ax}^2 + \text{bx} + \text{c}\) into \((px+q)(rx+s)\). | \(\text{x}^2 - 5\text{x} + 6 = (\text{x} - 2)(\text{x} - 3)\) |
Additional Information on Algebraic Factorization
Algebraic factorization is the process of breaking down a polynomial into a product of simpler polynomials. This is a fundamental skill in algebra used for simplifying expressions, solving equations, and analyzing functions.
Different techniques are used depending on the structure of the polynomial:
- Greatest Common Factor (GCF): Always look for a common factor first.
- Grouping: Useful for polynomials with four or more terms, as shown in this problem.
- Identities: Recognizing patterns like difference of squares, perfect square trinomials, sum/difference of cubes.
- Factoring Trinomials: Techniques like splitting the middle term or using the AC method for quadratics.
- Substitution: As demonstrated in the solution, substituting a part of the expression with a single variable can simplify complex polynomials into recognizable forms.
Mastering these techniques requires practice in recognizing the underlying structures within complex expressions.
Important Questions from Polynomials
If (x + y) 3+ 8 (x - y) 3= (3x + Ay) (3x 2+ Bxy + Cy 2), then the value of A + B + C is:
Given that x 8- 34x 4+ 1 = 0, x > 0. What is the value of (x 3+ x -3 )?
If \(x - \frac 3 x = 6,\; x \ne 0,\) then the value of \(\frac {x^4 - \frac {27}{x^2}}{x^2 - 3x - 3}\) is:
If \(x\left(3 - \frac 2 x\right) = \frac 3 x,\) then the value of \(x^3 - \frac 1 {x^3}\) is equal to:
The coefficient of x in (x – 3y) 3is:
