Examples of Factored Form in Algebra
The concept of Factored Form is essential in mathematics and helps in solving real-world and exam-level problems efficiently. Understanding how to write an algebraic expression or a polynomial in factored form makes finding solutions and simplifying calculations much easier, especially in algebraic problem-solving.
Understanding Factored Form
A factored form refers to rewriting an algebraic expression, especially polynomials, as a product of its simplest possible factors. This method is widely used in quadratic equations, polynomial factorization, and understanding the roots of equations. In the factored form, you can easily identify the solutions (roots or zeroes) of the equation.
Factored Form Definition and Formula
In simple terms, the factored form of a polynomial is when the expression is written as a product of factors, which can be constants, variables, or polynomials that cannot be factored further. For example:
The standard formula for the factored form of a quadratic equation is:
\( ax^2 + bx + c = a(x - r_1)(x - r_2) \)
Here’s a helpful table to understand factored form more clearly:
Factored Form Table
| Expression | Standard Form | Factored Form |
|---|---|---|
| Quadratic | \( x^2 - 5x + 6 \) | \( (x - 2)(x - 3) \) |
| Quadratic with GCD | \( 3x^2 - 6x + 12 \) | \( 3(x^2 - 2x + 4) \) |
| Difference of Squares | \( y^2 - 100 \) | \( (y + 10)(y - 10) \) |
This table shows common scenarios where factored form simplifies both calculation and interpretation of roots.
How to Convert to Factored Form: Step-by-Step Guide
Let's convert a quadratic equation into its factored form using stepwise reasoning:
1. Start with the equation: \( x^2 - 5x + 6 = 0 \)
2. Find two numbers that multiply to 6 (constant term) and add up to -5 (coefficient of \( x \)). In this case, -2 and -3.
3. Rewrite the equation: \( x^2 - 2x - 3x + 6 = 0 \)
4. Factor by grouping: \( x(x - 2) - 3(x - 2) = 0 \)
5. Factor the common term: \( (x - 2)(x - 3) = 0 \)
6. The equation is now in factored form. The solutions (roots) are \( x = 2 \) and \( x = 3 \).
Worked Example – Factoring a Quadratic
Let’s factor \( 12y^2 - 27 \):
1. Identify the GCD of coefficients. Both terms have a GCD of 3.
2. Factor GCD: \( 12y^2 - 27 = 3(4y^2 - 9) \)
3. Notice \( 4y^2 - 9 \) is a difference of squares: \( (2y)^2 - (3)^2 \)
4. Factor the difference: \( 3(2y + 3)(2y - 3) \)
5. Final answer: \( 12y^2 - 27 = 3(2y + 3)(2y - 3) \)
Practice Problems
- Write \( x^2 + 7x + 12 \) in factored form.
- Factor \( 5x^2 - 20 \).
- Express \( y^2 - 16 \) as a product of its factors.
- Find the factored form of \( x^2 - 4x - 12 \).
Factored Form vs Standard Form
| Aspect | Standard Form | Factored Form |
|---|---|---|
| Quadratic | \( x^2 + bx + c \) | \( (x + p)(x + q) \) |
| Directly Shows Roots? | No | Yes |
| Good for Graphing? | Sometimes | Yes |
Factored form makes it easy to see the roots or solutions directly, while standard form is useful for some calculations. Knowing both forms supports math skills in exams and problem-solving.
Uses & Applications of Factored Form
- Solving quadratic and polynomial equations quickly.
- Finding roots or zeroes for graphing parabolas or lines.
- Simplifying expressions to make calculations easier.
- Checking the solutions of algebraic problems.
- Understanding word problems in real-life contexts, such as area calculations or product sales.
Common Mistakes to Avoid
- Confusing factored form with standard or expanded form (e.g., not recognizing product format).
- Forgetting to factor completely (missing GCD or difference of squares).
- Incorrect grouping or splitting of terms in polynomials.
- Not checking if factors can be simplified further.
Real-World Applications
The concept of factored form appears in areas such as architecture, design, engineering, and science, especially when solving area, volume, or optimization problems. Vedantu helps students see how maths applies beyond the classroom, making factored form an important practical tool.
We explored the idea of Factored Form, how to apply it, solve related problems, and understand its real-life relevance. Practice more with Vedantu to build confidence in these concepts.
Related Vedantu Maths Resources
FAQs on How to Write an Expression in Factored Form
1. What does 'factored form' mean in mathematics?
The factored form of an expression or equation means it is written as a product of its factors. For example, for polynomials, factored form shows all possible prime factors, making it easier to solve or simplify the expression.
2. How do you convert an expression to factored form?
To convert an expression to its factored form:
1. Identify the greatest common factor (GCF) of all terms.
2. Factor out the GCF.
3. Apply any applicable factoring techniques such as difference of squares, trinomials, or grouping to further factor the remaining expression if possible.
3. What is the factored form of 3x + 24y?
The factored form of 3x + 24y is 3(x + 8y), where 3 is the GCF factored out from both terms.
4. How do you write a formula in factored form?
To write any formula in factored form, factor out all common terms or use suitable factoring methods so the expression is written as a product of its factors. For example, ax2 + bx + c can often be written as (x + p)(x + q) in factored form.
5. What are some examples of factored forms?
Examples of factored forms:
• x2 - 9 = (x - 3)(x + 3)
• 2x + 6 = 2(x + 3)
• x2 + 5x + 6 = (x + 2)(x + 3)
6. What is the factored form of a quadratic equation?
The factored form of a quadratic equation ax2 + bx + c is usually written as a(x - r1)(x - r2), where r1 and r2 are the roots or zeros of the equation.
7. How do you graph a quadratic function in factored form?
To graph a quadratic in factored form (y = a(x - r1)(x - r2)), plot the x-intercepts at r1 and r2, determine the direction using the sign of 'a', and find the vertex between the intercepts to complete the parabola.
8. What is the difference between standard form and factored form?
The standard form of a quadratic is ax2 + bx + c, while its factored form is a(x - r1)(x - r2). The factored form explicitly shows the roots or zeros of the equation, making it easier to solve for x-intercepts.
9. Can a polynomial always be written in factored form?
Every polynomial can be written in factored form using its prime factors. However, some polynomials have factors that cannot be simplified further over real numbers and may require complex numbers.
10. How do you use a factored form calculator?
A factored form calculator accepts a polynomial expression, applies factoring algorithms, and displays the expression as a product of factors. You simply enter the polynomial and select the factor option for step-by-step solutions.
11. What is the definition of factored form for a quadratic function?
The factored form of a quadratic function is written as f(x) = a(x – r1)(x – r2), where a ≠ 0, and r1, r2 are the function's zeros. It highlights the solutions easily.
12. How do you convert factored form to vertex form?
To convert a quadratic from factored form to vertex form:
1. Expand the factors into standard form.
2. Complete the square to rewrite in vertex form, which is f(x) = a(x – h)2 + k, where (h, k) is the vertex of the parabola.
