Chapter 11: Problem 24
Factor completely, by hand or by calculator. Check your results. The General Quadratic Trinomial. $$6 x^{2}-7 x+2$$
Short Answer
Expert verified
The factored form of the quadratic trinomial \(6x^2-7x+2\) is \((3x-2)(2x-1)\).
Step by step solution
01
Determine the Coefficients
Identify the coefficients of the quadratic trinomial. Here, the coefficients are: A=6 (leading coefficient), B=-7 (linear coefficient), and C=2 (constant term).
02
Find Two Numbers
Find two numbers that multiply to A*C (which is 12) and add up to B (which is -7). The numbers that satisfy these conditions are -3 and -4.
03
Rewrite the Middle Term
Rewrite the quadratic trinomial by breaking the middle term into two terms using the numbers found in Step 2: \(6x^{2}-3x-4x+2\).
04
Factor by Grouping
Group the terms into two pairs and factor each pair separately: \((6x^{2}-3x) + (-4x+2)\). Factoring out the common factors gives \(3x(2x-1) -2(2x-1)\).
05
Factor Out the Common Binomial Factor
Since both groups contain the common binomial factor \((2x-1)\), factor it out: \((3x-2)(2x-1)\).
06
Verify the Results
Check the factored form by multiplying the binomials to ensure it equals the original trinomial: \((3x-2)(2x-1) = 6x^2 - 3x - 4x + 2 = 6x^2 - 7x + 2\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Trinomial Coefficients
Understanding the role of coefficients in a quadratic trinomial is crucial for effective factoring. A quadratic trinomial, such as \(6 x^{2}-7 x+2\), can be identified by its distinct parts: the leading coefficient (A), the linear coefficient (B), and the constant term (C).
In our example, \(A=6\), \(B=-7\), and \(C=2\). These values are not just random numbers; they are the key that unlocks the factoring process. The leading coefficient, A, impacts the steepness of the parabola represented by the quadratic equation. The linear coefficient, B, influences the direction and the point at which the parabola crosses the y-axis. Lastly, the constant term, C, affects where the parabola touches the x-axis.
Knowing these coefficients allows you to search for specific numbers in the factoring process. In the provided exercise, the aim was to find two numbers that multiply together to give A*C - a product that will reveal possible numbers that can be used to split the middle term and move towards a factored form.
In our example, \(A=6\), \(B=-7\), and \(C=2\). These values are not just random numbers; they are the key that unlocks the factoring process. The leading coefficient, A, impacts the steepness of the parabola represented by the quadratic equation. The linear coefficient, B, influences the direction and the point at which the parabola crosses the y-axis. Lastly, the constant term, C, affects where the parabola touches the x-axis.
Knowing these coefficients allows you to search for specific numbers in the factoring process. In the provided exercise, the aim was to find two numbers that multiply together to give A*C - a product that will reveal possible numbers that can be used to split the middle term and move towards a factored form.
Factoring by Grouping
Factoring by grouping is a powerful technique when handling more complex quadratic trinomials. It involves rearranging and grouping terms in such a way that each group has a common factor that can be factored out.
Consider our quadratic trinomial \(6x^{2}-7x+2\). The middle term was split based on the numbers that were identified earlier, -3 and -4, which were then grouped to allow for the factoring process: \((6x^{2}-3x) + (-4x+2)\). Grouping the terms requires a keen eye for common factors within those groups. In our example, each group had a common factor that we could extract: \(3x\) from the first group and \(-2\) from the second.
After factoring out these common factors, what's left in each group should be a matching binomial, \((2x-1)\), which reveals that the original expression can be written as the product of two binomials, ultimately leading us to the completely factored form of the quadratic trinomial.
Consider our quadratic trinomial \(6x^{2}-7x+2\). The middle term was split based on the numbers that were identified earlier, -3 and -4, which were then grouped to allow for the factoring process: \((6x^{2}-3x) + (-4x+2)\). Grouping the terms requires a keen eye for common factors within those groups. In our example, each group had a common factor that we could extract: \(3x\) from the first group and \(-2\) from the second.
After factoring out these common factors, what's left in each group should be a matching binomial, \((2x-1)\), which reveals that the original expression can be written as the product of two binomials, ultimately leading us to the completely factored form of the quadratic trinomial.
Factoring Techniques
There are several factoring techniques, and each has its place depending on the form and complexity of the quadratic equation at hand. The most common techniques include finding a common factor, factoring by grouping, the difference of two squares, the square of a binomial, and the quadratic formula.
In the case of the exercise \(6x^{2}-7x+2\), we utilized both finding a common factor and the grouping method. These methods are often used in tandem to break down a trinomial into a product of binomials. It is important to understand when and how to apply each technique. For example, factoring out a greatest common factor should always be the first step, while the grouping method is immensely helpful when dealing with trinomials, where A is greater than 1.
Each technique requires a different approach and understanding of the relationship between the coefficients. Mastering these techniques not only helps in factoring quadratics but also in understanding the deeper structures of algebraic expressions and their properties.
In the case of the exercise \(6x^{2}-7x+2\), we utilized both finding a common factor and the grouping method. These methods are often used in tandem to break down a trinomial into a product of binomials. It is important to understand when and how to apply each technique. For example, factoring out a greatest common factor should always be the first step, while the grouping method is immensely helpful when dealing with trinomials, where A is greater than 1.
Each technique requires a different approach and understanding of the relationship between the coefficients. Mastering these techniques not only helps in factoring quadratics but also in understanding the deeper structures of algebraic expressions and their properties.
