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12.2.4. Factors of the form (x + a)(x + b)
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Today, we'll explore how to factor expressions like x² + 5x + 6. Can anyone remind me what we mean by 'factoring'?
Isn't it when we write something as a product of its factors?
Exactly! When we say factor, we’re talking about breaking down that expression into simpler components. Now, let's see this directly related to our identity. For our expression, if we think of it in terms of the identity (x + a)(x + b), what must a and b be?
The product ab is 6, and a + b is 5!
Correct! Those two conditions will help us find the right pair. Could anyone suggest pairs of numbers that multiply to 6?
2 and 3!
Yes! Not only do 2 and 3 multiply to 6, but they also add up to 5. Let’s write down our factored form: (x + 2)(x + 3).
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Now, let’s take a look at an expression like z² - 4z - 12. Any thoughts on how we might approach this?
I think we have to find numbers that multiply to -12?
Correct! And we also need to ensure that they add up to -4. So, what pairs can we try?
How about -6 and 2? They multiply to -12 and sum to -4!
Excellent! So we can express this as (z - 6)(z + 2).
This is getting clearer. We look for pairs based on their product and sum!
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Create a free accountLet’s practice! Factor the expression x² - 10x + 21. Can anyone start us off?
The product of ab should be 21, right? And the sum -10?
Right! What pairs can yield a product of 21 and a sum of 10?
It would be -3 and -7!
Exactly! Thus, we factor it to (x - 3)(x - 7). Nice work! This method works for any similar quadratic expressions.
Overview
Short Summary
This section discusses how to factor expressions in one variable, specifically those that can be expressed in the form of (xxxxx + aaaaa)(xxxxx + bbbbb), including strategies to find the correct factors.
Medium Summary
The section explores the process of factorizing quadratic expressions such as x² + 5x + 6 using the identity (x + a)(x + b) = x² + (a + b)x + ab. It highlights how to identify coefficients and derive the necessary factors to simplify the expressions effectively.
Detailed Summary
Factors of the form (x + a)(x + b)
This section elaborates on the procedures to factorize algebraic expressions composed of a single variable, particularly those expressed in the format equivalent to the quadratic identity (x + a)(x + b) = x² + (a + b)x + ab. The focus lies on identifying relevant coefficients from the standard expanded form and matching them to the factors of the constant term (ab).
For instance, consider the expression x² + 5x + 6. Here, the product ab equals 6, and the sum a + b must equal 5. Through the exploration of different factors, students can discover that using 2 and 3 satisfies both conditions—yielding the factorized form (x + 2)(x + 3). This section also implies that for expressions with negative or varied coefficients, a careful approach is required to factor them systematically.
Example :
Factorise
Solution: If we compare the R.H.S. of Identity (IV) with , we find , and . From this, we must find and . The factors then will be .
If , it means that and are factors of 10. Let us take , . For these values , and this choice is correct.
Let us try , . For this, which is not exactly required.
The factorised form of this given expression is then .
Reference YouTube Videos
Key Concepts
Core takeaways and short definitions to help you quickly recall the key ideas from this section.
Quadratic Factorization: The process of breaking down quadratic expressions into factors of the form (x + a)(x + b).
Product and Sum: For factors a and b, ab is the constant term and a + b is the coefficient of x in a quadratic expression.
Identity Application: Identifying appropriate coefficients to apply the quadratic factorization identity.
Examples
Memory Aids
Interactive tools to help you remember key concepts
Stories
Flash Cards
Glossary
Quadratic Expression
An algebraic expression of the form ax² + bx + c.
Factoring
The process of breaking down an expression into a product of its factors.
Identity
A mathematical statement that holds true for all values of its variable.
Coefficients
Numerical or constant quantity placed before a variable in an algebraic expression.