Problem 44 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The expression factors to \((3t + 2)(-2t + 3)\).

Step 1: Arrange Terms in Standard Form

Start by writing the expression in standard quadratic form: \[-6t^2 + 5t + 6\].This arranges the terms in order of decreasing powers of \(t\).

Step 2: Identify Coefficients

Identify the coefficients of the quadratic equation:\(a = -6\), \(b = 5\), \(c = 6\).

Step 3: Find Two Numbers

Find two numbers that multiply to \(a \times c\) (which is \(-6 \times 6 = -36\)) and add to \(b\) (which is \(5\)).These numbers are \(-4\) and \(9\) because \(-4 \times 9 = -36\) and \(-4 + 9 = 5\).

Step 4: Rewrite Middle Term

Rewrite the middle term (\(5t\)) of the expression using the two numbers found:\[-6t^2 - 4t + 9t + 6\].

Step 5: Factor by Grouping

Group terms in pairs and factor out the greatest common factor (GCF) from each group:\[-2t(3t + 2) + 3(3t + 2)\].

Step 6: Factor Out the Common Binomial

Notice that \((3t + 2)\) is common in both terms, so factor it out:\[(3t + 2)(-2t + 3)\].

Key Concepts

Quadratic Equations

In a nutshell, quadratic equations are polynomial equations of degree two. They typically take the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients. The most prominent feature of these equations is their parabolic shape when graphed, known as a parabola.
Quadratics can have different numbers of solutions or roots:

• A parabola may intersect the x-axis at two points, indicating two real solutions.
• It might just touch the x-axis at one point, representing one real solution or double root.
• In some cases, it might not cross the x-axis at all, implying two complex solutions.

To solve quadratic equations, we often use factoring, which simplifies the expression into a product of simpler expressions. This can be done through methods such as factoring by grouping, which we will cover in the next section.

Factoring by Grouping

Factoring by grouping is a handy method for breaking down quadratic expressions, especially when the coefficient of the second-degree term (\(a\)) is not 1. This technique allows us to transform a complex expression into simpler binomials.
When using this method, follow these steps:

• First, arrange the equation in the standard quadratic form, \(ax^2 + bx + c\).
• Identify two numbers that multiply to \(ac\) and add to \(b\). This is a pivotal step.
• Rewrite the middle term with these two numbers, transforming it into four terms.
• Group the terms in pairs and factor out the greatest common factor from each group.
• Finally, factor the equation by taking out the common binomial.

Through grouping, we simplify the way to find factors, enabling us to solve quadratic equations more efficiently.

Polynomial Factorization

Polynomial factorization involves representing a polynomial as a product of its factors, which are of lower degrees than the polynomial itself. This is a crucial step in simplifying polynomials and finding their roots.
Every polynomial can be rewritten in terms of linear factors if it is factorizable. For quadratics, this means converting it into a product of two binomials.
For example, given \(-6t^2 + 5t + 6\), the factorization process involves finding and factoring out elements like \(3t + 2\) and \(-2t + 3\).

• Identify possible factors by examining the product of the first and last terms, \(ac\).
• Look for numbers that add to the middle coefficient \(b\).
• Rewrite the quadratic as a product of two binomials.

Factorization is not only vital for solving equations but also helps in graphing and other algebraic processes.