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Chapter 0: Problem 44

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

Short Answer

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

Step by step solution

01

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 \(t\).
02

Identify Coefficients

Identify the coefficients of the quadratic equation:\(a = -6\), \(b = 5\), \(c = 6\).
03

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 \(-4 + 9 = 5\).
04

Rewrite Middle Term

Rewrite the middle term (\(5t\)) of the expression using the two numbers found:\[-6t^2 - 4t + 9t + 6\].
05

Factor by Grouping

Group terms in pairs and factor out the greatest common factor (GCF) from each group:\[-2t(3t + 2) + 3(3t + 2)\].
06

Factor Out the Common Binomial

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Complete the following tables. What happens to the \(n\) th root of 2 as \(n\) gets large? What about the \(n\) th root of \(\frac{1}{2} ?\) \(\begin{array}{|c|c|}\hline n & {2^{1 / n}} \\ \hline 1 & {} \\ {2} & {} \\\ {5} \\ {10} \\ {100} & {} \\ \hline\end{array}\) \(\begin{array}{|c|c|}\hline n & {\left(\frac{1}{2}\right)^{1 / n}} \\ \hline 1 & {} \\ {2} & {} \\ {5} & {} \\ {10} \\ {100} & {} \\ \hline\end{array}\) Construct a similar table for \(n^{1 / n} .\) What happens to the \(n\) th root of \(n\) as \(n\) gets large?

Volume of the Oceans The average ocean depth is \(3.7 \times 10^{3} \mathrm{m},\) and the area of the oceans is \(3.6 \times 10^{14} \mathrm{m}^{2}\) . What is the total volume of the ocean in liters? (One cubic meter contains 1000 liters.)

\(89-96\) m State whether the given equation is true for all values of the variables. (Disregard any value that makes a denominator zero.) $$ \frac{x+1}{y+1}=\frac{x}{y} $$

\(83-88=\) Rationalize the numerator. $$ \frac{1-\sqrt{5}}{3} $$

\(55-64=\) Simplify the compound fractional expression. $$ 1+\frac{1}{1+\frac{1}{1+x}} $$
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