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Chapter 6: Problem 9

Factor each trinomial. $$ m^{2}-11 m+60 $$

Short Answer

Expert verified
The factors are \((m - 5)(m - 6)\).

Step by step solution

01

- Identify the trinomial structure

Consider the given trinomial in standard form: \(m^2 - 11m + 60\). This is a quadratic equation in the form \(ax^2 + bx + c\). Here, \(a = 1\), \(b = -11\), and \(c = 60\).
02

- Find two numbers that multiply to \(c\) and add to \(b\)

We need to find two numbers that multiply to 60 (the constant term \(c\)) and add to -11 (the coefficient of the middle term \(b\)). These two numbers are -5 and -6 since \(-5 * -6 = 30\) and \(-5 + -6 = -11\).
03

- Rewrite the middle term

Rewrite \(-11m\) as \(-5m - 6m\) in the trinomial. So, the trinomial becomes \(m^2 - 5m - 6m + 60\).
04

- Factor by grouping

Group the terms in pairs: \((m^2 - 5m) + (-6m + 60)\). Factor out the greatest common factor (GCF) from each pair: \(m(m - 5) - 6(m - 5)\).
05

- Factor out the common binomial

Notice the common factor \((m - 5)\). Factor it out, resulting in \((m - 5)(m - 6)\).

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Key Concepts

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

Quadratic Equations
A quadratic equation is a second-degree polynomial, which means its highest exponent is squared. It generally takes the form ewline $$ax^2 + bx + c = 0$$. In this equation,
  • *\(a\), \(b\), and \(c\) are constants.*
  • * \(x\) is the variable.

The solution to a quadratic equation means finding the values of \(x\) that make the equation true.
These equations can be solved using various methods like factoring, completing the square, or using the quadratic formula.

In the original exercise, the trinomial \(m^2 - 11m + 60\) is a quadratic equation in the form where \(a = 1\), \(b = -11\), and \(c = 60\). The goal is to use factoring to solve it.
Factoring by Grouping
Factoring by grouping is a technique used to simplify polynomial expressions by grouping terms with common factors.

From the original exercise:* In \(m^2 - 11m + 60\), the key is to rewrite \(-11m\) as \(-5m - 6m\), splitting the middle term so it fits our needs.*
Next, you group the terms: $$m^2 - 5m - 6m + 60$$ becomes $$(m^2 - 5m) + (-6m + 60)$$.

The final step is to factor out the Greatest Common Factor (GCF) from each group:
$$m(m - 5) - 6(m - 5)$$.
Notice the common term \((m - 5)\) in each group.This technique is especially useful when direct factoring seems difficult.
Greatest Common Factor
The Greatest Common Factor (GCF) is the highest number or expression that divides exactly into two or more numbers or terms. In factoring problems, finding the GCF can simplify expressions and make them easier to work with.

In step 4 of the original solution:
Terms in pairs \((m^2 - 5m)\) and \((-6m + 60)\) were grouped.

Notice how:

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Most popular questions from this chapter

In some cases, the method of factoring by grouping can be combined with the methods of special factoring discussed in this section. Consider this example. $$ \begin{aligned} 8 x^{3} &+4 x^{2}+27 y^{3}-9 y^{2} \\ &=\left(8 x^{3}+27 y^{3}\right)+\left(4 x^{2}-9 y^{2}\right) \end{aligned} $$ $$ \begin{aligned} &=(2 x+3 y)\left(4 x^{2}-6 x y+9 y^{2}\right)+(2 x+3 y)(2 x-3 y)\\\ &=(2 x+3 y)\left[\left(4 x^{2}-6 x y+9 y^{2}\right)+(2 x-3 y)\right]\\\ &=(2 x+3 y)\left(4 x^{2}-6 x y+9 y^{2}+2 x-3 y\right) \end{aligned} $$ In problems such as this, how we choose to group in the first step is essential to factoring correctly. If we reach a "dead end," then we should group differently and try again. Use the method just described to factor each polynomial. $$ y^{4}+y^{3}+y+1 $$

Factor each trinomial. $$ 10 x^{2}+3 x-18 $$

If a baseball is dropped from a helicopter \(625 \mathrm{ft}\) above the ground, then its distance in feet from the ground \(t\) seconds later is a function defined by $$f(t)=-16 t^{2}+625$$ How long after it is dropped will it hit the ground?

Factor each trinomial. $$ 24 x^{2}+42 x+15 $$

Factor each trinomial. $$ 4 p^{3}+24 p^{2}-64 p $$
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