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Chapter 6: Problem 39

Factor Does the trinomial \(3 x^{2}+13 x-16\) have a common monomial factor?

Short Answer

Expert verified
No, the trinomial \(3x^{2}+13x-16\) does not have a common monomial factor apart from 1.

Step by step solution

01

Identify the trinomial

The trinomial given in the problem is \(3x^{2}+13x-16\).
02

Check for common factors in all the coefficients

A common factor would be any number or variable that divides evenly into the coefficients of all terms of the trinomial. The coefficients in this case are 3, 13, and -16. However, 3, 13, and -16 have no common factors other than 1. Hence, we can conclude that there are no common monomial factors for the given trinomial.
03

Confirm the results

The only common factor of 3, 13, and -16 is 1, which does not help in simplifying the trinomial or expressing it in factored form. This confirms our previous findings.

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Key Concepts

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

Understanding Algebraic Expressions
Algebraic expressions are combinations of variables, constants, and arithmetic operations like addition, subtraction, multiplication, and division. They are the building blocks of algebra. An example of an algebraic expression is \(3x^2 + 13x - 16\), which is a trinomial as it consists of three terms.
Understanding the structure of algebraic expressions is crucial for factoring, simplifying, and solving equations.
When dealing with algebraic expressions, we often seek to write them in different forms, such as factored form, in order to solve equations or simplify expressions.
Exploring Polynomials
Polynomials are a type of algebraic expression that consists of multiple terms. These terms are made up of variables raised to whole-number exponents and their coefficients. A polynomial can have one variable or multiple variables, and the degree of the polynomial is determined by the highest power of the variable.
Trinomials, like \(3x^2 + 13x - 16\), are a specific type of polynomial with exactly three terms.
  • One of the terms will usually be a quadratic term, like \(3x^2\).
  • The middle term is a linear term, like \(13x\).
  • The last term is a constant, like \(-16\).
Factoring polynomials, especially trinomials, involves finding expressions that, when multiplied together, give the original polynomial. This process often requires looking for patterns or using techniques like the distributive property to simplify the expression.
The Greatest Common Factor in Polynomials
The Greatest Common Factor (GCF) is the largest number or expression that divides all terms in a polynomial with no remainder. Identifying the GCF is a fundamental step in the process of factoring polynomials. It allows us to simplify expressions by "pulling out" common factors from multiple terms.
For the trinomial \(3x^2 + 13x - 16\), the coefficients are 3, 13, and -16. To find the GCF:
  • We examine the numbers to check if there is a common divisor.
  • If no number except 1 divides all coefficients, like in this trinomial, then we say it does not have a common monomial factor.
In cases where a trinomial does not have a GCF other than 1, it is as simplified as it can be in terms of common factors. Understanding why a GCF might not exist is crucial for confirming the fully factored form of a polynomial.

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Most popular questions from this chapter

Is the equation \(2 x^{2}+18 x=0\) in general form? Explain.

Determine whether the statement is true or false. Justify your answer. If \(a\) and \(b\) are nonzero real numbers, then the solutions of the equations \(a x^{2}+b x=0\) are \(x=0\) and \(x=-\frac{b}{a}\).

In Exercise 83 and 84, write the polynomial as the difference of two squares. Use the result to factor the polynomial completely. $$ \begin{aligned} &x^{2}+8 x+12=\left(x^{2}+8 x+16\right)-4\\\ &= \end{aligned} $$

Factor the trinomial. $$ 15 x^{2}-28 x+12 $$

Find two consecutive positive even integers whose product is 440 .
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