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Chapter 6: Problem 3

For Problems \(1-50\), factor each of the trinomials completely. Indicate any that are not factorable using integers. (Objective 1) $$ 6 x^{2}+19 x+10 $$

Short Answer

Expert verified
The trinomial \(6x^2 + 19x + 10\) factors to \((2x + 5)(3x + 2)\).

Step by step solution

01

Identify and Set Up the Trinomial

The problem asks us to factor the trinomial \(6x^2 + 19x + 10\). This type of trinomial is of the form \(ax^2 + bx + c\), where \(a = 6\), \(b = 19\), and \(c = 10\). The goal is to express it as a product of two binomials.
02

Find Two Numbers with a Special Property

Look for two numbers that multiply to \(a \cdot c = 6 \cdot 10 = 60\) and add to \(b = 19\). After testing various factor pairs of 60, we find that \(4\) and \(15\) are our two numbers, because \(4 + 15 = 19\) and \(4 \times 15 = 60\).
03

Rewrite the Middle Term

Use these numbers to rewrite the middle term of the trinomial as two separate terms: \(19x = 4x + 15x\). Thus, our expression becomes \(6x^2 + 4x + 15x + 10\).
04

Group and Factor by Grouping

Group the terms into pairs: \((6x^2 + 4x) + (15x + 10)\). Factor each pair separately: from \(6x^2 + 4x\), factor out \(2x\) to get \(2x(3x + 2)\). From \(15x + 10\), factor out \(5\) to get \(5(3x + 2)\).
05

Factor Out the Common Binomial

Notice that both groups contain the common binomial \((3x + 2)\). Factor this out, yielding \((2x + 5)(3x + 2)\).
06

Verify the Factorization

To ensure the factorization is correct, multiply the binomials: \((2x + 5)(3x + 2) = 6x^2 + 4x + 15x + 10 = 6x^2 + 19x + 10\). The original expression is obtained, confirming the factorization is accurate.

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Key Concepts

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

Binomial Factorization
Binomial factorization is a process used in algebra to break down expressions into more manageable pieces, called binomials. A binomial is simply a mathematical expression containing two terms connected by either a plus or a minus sign. Factoring a trinomial into binomials involves finding two expressions that multiply together to form the original trinomial.
  • In our exercise, we started with the trinomial expression: \(6x^2 + 19x + 10\).
  • The objective was to find two binomials such that when multiplied, they recreate this trinomial.
  • This involves identifying two numbers that meet a specific condition related to the coefficients.
Learning binomial factorization is crucial as it simplifies complex expressions, making them easier to solve, especially in quadratic equations.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operations that represent a particular value or set of values. Understanding how to factor these expressions is essential for solving equations and simplifying mathematical models.
  • Algebraic expressions can range from simple, like \(3x + 2\) to more complex forms such as trinomial in the exercise: \(6x^2 + 19x + 10\).
  • The expression consists of coefficients (numbers), variables (such as \(x\)), and constants.
When factoring these expressions, we look for ways to express them as products of simpler expressions. This helps in simplifying problems and solving for unknowns more efficiently.
Quadratic Trinomials
Quadratic trinomials are a specific type of algebraic expressions with three terms, generally in the form \(ax^2 + bx + c\). Each part has a role:
  • The term \(ax^2\) is known as the quadratic term.
  • \(bx\) is the linear term.
  • \(c\) represents the constant term.
In factoring, the key is to transform the trinomial into a product of two binomials. The Step by Step Solution shows how to do this by finding two numbers that multiply to \(a \cdot c\) and add to \(b\). With \(6x^2 + 19x + 10\), we transformed it into \((2x + 5)(3x + 2)\).
Recognizing patterns in these trinomials and understanding how each coefficient interplays is crucial for successful factorization.

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Most popular questions from this chapter

For Problems \(71-88\), set up an equation and solve each problem. (Objective 4) Find two numbers whose product is \(-1\). One of the numbers is three more than twice the other number.

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