Problem 48 - Algebra and Trigonometry

Question

Factor the expression completely. $$ 6+5 t-6 t^{2} $$

Step-by-Step Solution

Answer

The completely factored form is \((2t - 3)(3t - 2)\).

Step 1: Arrange in Standard Form

The given expression is \(6 + 5t - 6t^2\). First, arrange the terms in descending order of the power of \(t\). The expression becomes: \(-6t^2 + 5t + 6\).

Step 2: Factor Out the Greatest Common Factor

In this step, we look for the greatest common factor (GCF) of all the coefficients. Here, the GCF is 1, so we proceed to factor the quadratic expression: \(-6t^2 + 5t + 6\).

Step 3: Apply the AC Method

To factor the quadratic expression \(-6t^2 + 5t + 6\), use the AC method. Multiply the coefficient of \(t^2\), which is \(-6\), by the constant term \(c\), which is \(6\). Thus, AC = \(-6 \times 6 = -36\). We need two numbers that multiply to \(-36\) and add to \(5\). These numbers are \(9\) and \(-4\).

Step 4: Rewrite the Middle Term

Rewrite the expression \(-6t^2 + 5t + 6\) by using the two numbers found in Step 3: \(-6t^2 + 9t - 4t + 6\).

Step 5: Factor by Grouping

Group the terms: \((-6t^2 + 9t) + (-4t + 6)\). Factor out the common factor in each group: \(3t(-2t + 3) - 2(-2t + 3)\). Notice \(-2t + 3\) is common in both groups. Factor it out: \((3t - 2)(-2t + 3)\).

Step 6: Simplify the Factors

Adjust any negative signs and simplify the expression if necessary: \((3t - 2)(-2t + 3)\), since the problem requests complete factorization without leading negatives, rewrite the factorization as: \((2t - 3)(3t - 2)\).

Step 7: Verify the Factorization

Expand \((2t - 3)(3t - 2)\) to verify it equals the original expression. The expansion gives \(6t^2 - 4t - 9t + 6 = -6t^2 + 5t + 6\), confirming the factorization is correct.

Key Concepts

Quadratic Expression

A quadratic expression is an algebraic expression of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The highest power of the variable, typically \( x \), is 2. Quadratics are common in algebra and have unique properties that allow them to be factored into simpler expressions or equations.
Quadratic expressions can represent various real-world scenarios. Understanding them is essential for solving quadratic equations, which are equations set equal to zero.
When working with quadratic expressions, first write them in standard form, rearranging terms in descending order of degree. This will help identify coefficients \( a \), \( b \), and \( c \), which are crucial for applying methods like factoring.

AC Method

The AC method is a technique used to factor quadratic expressions, particularly when traditional factoring is not immediately straightforward. It's great for expressions where \( a eq 1 \) in the standard form \( ax^2 + bx + c \). To use the AC method:- Multiply \( a \) (the coefficient of \( x^2 \)) by \( c \) (the constant term).- Find two numbers that multiply to the product \( ac \) and add up to \( b \).- Split the middle term \( bx \) using these two numbers, transforming the quadratic into a four-term expression.In the example exercise, the product of \( -6 \) and \( 6 \) is \( -36 \). The numbers \( 9 \) and \( -4 \) multiply to give \( -36 \) and add up to \( 5 \). This crucial step helps transform the expression for further factoring.

Factoring by Grouping

Factoring by grouping is a strategy used to factor polynomials, especially after using the AC method. Once the quadratic expression is rewritten with four terms:- Group the terms into two pairs.- Factor out the greatest common factor from each pair.- If done correctly, a common binomial factor will appear in both groups.For example, with the expression \(-6t^2 + 9t - 4t + 6\):
- Group as \((-6t^2 + 9t) + (-4t + 6)\).- Factor out common factors to get \(3t(-2t + 3) - 2(-2t + 3)\).- The common factor, \(-2t + 3\), allows further factoring to get \((2t - 3)(3t - 2)\).This method efficiently breaks down complex quadratic expressions into simpler, easily manageable parts.

Greatest Common Factor

The greatest common factor (GCF) is the largest factor shared by all terms in a polynomial. It's an essential concept in factoring, as it simplifies expressions and polynomials for easier manipulation and solving.
To find the GCF, identify the highest common number and variables shared by each term. Although sometimes the GCF is simply 1, as in our exercise, acknowledging it ensures no simplifications have been overlooked before applying other methods like the AC method or grouping.
Using the GCF as a starting point, the terms of a polynomial become easier to factor, especially when combined with additional factoring techniques.