Problem 43 - Algebra and Trigonometry

Question

31–76 ? Factor the expression completely. $$ 6 x^{2}-5 x-6 $$

Step-by-Step Solution

Answer

The expression \( 6x^2 - 5x - 6 \) factors to \((2x - 3)(3x + 2)\).

Step 1: Identify the form of the quadratic

The given expression is a quadratic trinomial. It is of the form \( ax^2 + bx + c \) where \( a = 6 \), \( b = -5 \), and \( c = -6 \). We will factor this trinomial.

Step 2: Find two numbers that multiply to ac and add to b

Calculate \( ac = 6 \times -6 = -36 \). We need two numbers whose product is \(-36\) and sum is \(-5\). These numbers are \(4\) and \(-9\) since \(4 + (-9) = -5\) and \(4 \cdot (-9) = -36\).

Step 3: Rewrite the middle term using the two numbers found

Rewrite \(-5x\) as \(4x - 9x\) to transform the expression into: \[ 6x^2 + 4x - 9x - 6 \]

Step 4: Factor by grouping

Group the terms as follows: \( (6x^2 + 4x) + (-9x - 6) \). Factor out the greatest common factor in each group: \( 2x(3x + 2) - 3(3x + 2) \).

Step 5: Factor out the common binomial

Notice that \((3x + 2)\) is a common factor: \[ (2x - 3)(3x + 2) \] So, the expression is fully factored.

Key Concepts

Quadratic Trinomial

A quadratic trinomial is an algebraic expression consisting of three terms. It is of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). This type of polynomial stands out because the highest degree is always two, hence the name 'quadratic.' In this specific problem, \( 6x^2 - 5x - 6 \) is our quadratic trinomial.

When we factor a quadratic trinomial, we are essentially expressing it as a product of two binomials. This means finding two expressions that, when multiplied, will give us the original trinomial.

• The first term \( ax^2 \) comes from multiplying the first terms of each binomial.
• The last term \( c \) results from multiplying the last terms.
• The middle term \( bx \) is derived from adding the outer and inner products.

Understanding how these terms interact with each other is key to mastering the skill of factoring quadratics.

Factoring by Grouping

Factoring by grouping is a method used to factor more complex expressions, including quadratic trinomials. This technique is particularly useful when the trinomial cannot be factored easily with simpler forms. In the exercise \( 6x^2 - 5x - 6 \), after rewriting the middle term using the suitable numbers \(4\) and \(-9\), we get the expression \( 6x^2 + 4x - 9x - 6 \).

By factoring by grouping, we split this expression into two parts: \((6x^2 + 4x)\) and \((-9x - 6)\). Each part can be simplified by factoring out the greatest common factor:
* For \(6x^2 + 4x\), the greatest common factor is \(2x\), resulting in \(2x(3x + 2)\).
* For \(-9x - 6\), the greatest common factor is \(-3\), simplifying it to \(-3(3x + 2)\).

Finally, notice the expression now includes a common factor of \(3x + 2\), which we can factor out. This strategic grouping helps convert more challenging trinomials into manageable binomials.

Greatest Common Factor

The Greatest Common Factor (GCF) is crucial in simplifying expressions. It is the largest factor that two or more numbers have in common. In the context of factoring polynomials, identifying the GCF of terms allows us to simplify expressions by breaking them into simpler, more usable parts.

In the example, when we group \((6x^2 + 4x)\) and \((-9x - 6)\), we seek out the GCF in each group:

• The GCF of \(6x^2 + 4x\) is \(2x\). Here, both terms are divisible by \(2x\).
• For \(-9x - 6\), the GCF is \(-3\). This will allow us to consistently factor through negative terms.

By factoring these GCFs out, we not only simplify each group but also reveal any common binomial factors. This step is indispensable for transforming a polynomial like \(6x^2 - 5x - 6\) into the product \((2x - 3)(3x + 2)\), completing the factorization process.