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Chapter 7: Problem 50

Factor out the greatest common factor. Be sure to check your answer. Factor out \(-q\) from \(-10 q^{3}-4 q^{2}+q\)

Short Answer

Expert verified
\( -2q(5q^{2}+2q-\dfrac{1}{2}) \)

Step by step solution

01

Identify the GCF of coefficients and variables

To find the GCF of coefficients, we'll compare the coefficients of each term in the given expression: \( -10 \), \( -4 \), and \( 1 \) The GCF of these coefficients is 2. Now, we need to find the GCF for variables. Since each term has a common variable q, we'll take the lowest exponent of q which is 1. Therefore, the GCF of variables is \( q \). The GCF of the expression is -2q.
02

Factor out the GCF

Now, we will factor out -2q from the given expression. \(( -2q)\) (first term \( 梅 \) GCF, second term \( 梅 \) GCF, third term \( 梅 \) GCF) \((-2q)[ \frac{-10q^{3}}{-2q} + \frac{-4q^{2}}{-2q} + \frac{q}{-2q} ] \)
03

Simplify the expression

After factoring out -2q, we now simplify the expression inside the brackets. \((-2q)(5q^{2} +2q - \dfrac{1}{2} )\) The final factored expression is: \( -2q(5q^{2}+2q-\dfrac{1}{2}) \)

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Key Concepts

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

Greatest Common Factor
The greatest common factor (GCF) is a fundamental concept in mathematics, especially when working with algebraic expressions. It refers to the largest factor that divides each term in a set of values without leaving a remainder. Finding the GCF involves identifying both the numerical and variable parts common to each term, as seen in the exercise where we factored the expression \(-10q^3 - 4q^2 + q\).
  • First, identify the GCF of the coefficients: here, the coefficients were \(-10\), \(-4\), and \(1\). The largest number that divides each of these is \(2\).
  • Next, look for common variables: in this case, each term contains the variable \(q\). The smallest exponent among these is 1, indicating that just \(q\) should be part of the GCF.
Combining these, the GCF of the expression was \(-2q\), which was factored out to simplify it further. Understanding and identifying the GCF is crucial to making algebraic manipulations more straightforward and solving problems efficiently.
Polynomials
Polynomials are expressions made up of variables and coefficients, linked together by addition, subtraction, and multiplication. No division by a variable is allowed in a polynomial. These expressions can range from simple to highly complex, but all have several defining characteristics. In the example of \(-10q^3 - 4q^2 + q\), it鈥檚 a polynomial consisting of three terms.
  • The degree of the polynomial is determined by the highest power of the variable, which in this case is 3 from the term \(-10q^3\).
  • Polynomials can be classified by their number of terms: a single-term expression is a monomial, a two-term is a binomial, and three or more terms make up a trinomial (as seen here).
  • Simplifying polynomials often involves factoring them, finding GCFs, or using other algebraic techniques to make them more manageable or to solve equations.
Understanding the nature of polynomials and their structure is essential when working with algebraic expressions and tackling more advanced algebra problems.
Algebraic Expressions
An algebraic expression is a mathematical phrase that combines numbers, variables, and operators. It represents a value and can range from simple expressions like \(x + 2\) to more complex ones like \(-10q^3 - 4q^2 + q\). These expressions are the building blocks of algebra, enabling you to represent real-world situations or to solve mathematical problems.
  • Expressions may not have an "equals" sign, distinguishing them from equations.
  • They can have constants, like \(-10\) or \(-4\), variables like \(q\), and coefficients that multiply the variables.
  • Simplifying algebraic expressions often involves combining like terms, factoring, or applying properties of operations such as the distributive property.
By breaking down algebraic expressions into their fundamental parts, you can better manipulate and understand them, crucial for solving equations and graphing functions.

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Most popular questions from this chapter

Factor completely. You may need to begin by taking out the GCF first or by rearranging terms. $$7 p q+28 q+14 p+56$$

Organizers of fireworks shows use quadratic and linear equations to help them design their programs. Shells contain the chemicals which produce the bursts we see in the sky. At a fireworks show the shells are shot from mortars and when the chemicals inside the shells ignite they explode, producing the brilliant bursts we see in the night sky. Shell size determines how high a shell will travel before exploding and how big its burst will be when it does explode. Large shell sizes go higher and produce larger bursts than small shell sizes. Pyrotechnicians take these factors into account when designing the shows so that they can determine the size of the safe zone for spectators and so that shows can be synchronized with music. At a fireworks show, a 3 -in. shell is shot from a mortar at an angle of \(75^{\circ} .\) The height, \(y\) (in feet), of the shell \(t\) sec after being shot from the mortar is given by the quadratic equation $$y=-16 t^{2}+144 t$$ and the horizontal distance of the shell from the mortar, \(x\) (in feet), is given by the linear equation $$x=39 t$$ a) How high is the shell after 3 sec? b) What is the shell's horizontal distance from the mortar after 3 sec? c) The maximum height is reached when the shell explodes. How high is the shell when it bursts after 4.5 sec? d) What is the shell's horizontal distance from its launching point when it explodes? (Round to the nearest foot.) When a 10 -in. shell is shot from a mortar at an angle of \(75^{\circ},\) the height, \(y\) (in feet), of the shell \(t\) sec after being shot from the mortar is given by $$y=-16 t^{2}+264 t$$and the horizontal distance of the shell from the mortar, \(x\) (in feet), is given by$$x=71 t$$ e) How high is the shell after 3 sec? f) Find the shell's horizontal distance from the mortar after 3 sec. g) The shell explodes after 8.25 sec. What is its height when it bursts? h) What is the shell's horizontal distance from its launching point when it explodes? (Round to the nearest foot.) i) Compare your answers to a) and e). What is the difference in their heights after 3 sec? j) Compare your answers to c) and g.). What is the difference in the shells' heights when they burst? k) Assuming that the technicians timed the firings of the 3-in. shell and the 10 -in. shell so that they exploded at the same time, how far apart would their respective mortars need to be so that the 10 -in. shell would burst directly above the 3 -in. shell?

Write an equation and solve. A 13 -ft ladder is leaning against a wall. The distance from the top of the ladder to the bottom of the wall is \(7 \mathrm{ft}\) more than the distance from the bottom of the ladder to the wall. Find the distance from the bottom of the ladder to the wall.

Factor completely. $$k^{4}-81$$

Factor completely. $$144 m^{2}-n^{4}$$
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