Problem 47 - Beginning and Intermediate Algebra

Question

Factor out the greatest common factor. Be sure to check your answer. Factor out \(-5 t^{2}\) from \(-5 t^{3}+10 t^{2}\)

Step-by-Step Solution

Answer

The short answer is: \(-5t^2(-t + 2)\)

Step 1: Identify the GCF

To identify the GCF, we look at both the coefficients and the variables in the polynomial \(-5t^3 + 10t^2\). for the coefficients: The GCF of \(-5\) and \(10\) is \(5\). Since this is a case of factoring out, we take the negative GCF which is \(-5\). for the variables: The GCF of the powers of \(t\) is the lowest power, which is \(t^2\). So the GCF is \(-5t^2\).

Step 2: Factor out the GCF

We will now factor out the GCF \(-5t^2\) from the polynomial \(-5t^3 + 10t^2\). \(-5t^2(-t + 2)\)

Step 3: Check the answer

To check our answer, we can multiply the GCF with the factored polynomial to see if we get the original polynomial. \(-5t^2(-t + 2) = -5t^2\times(-t) + (-5t^2)\times 2\) \(= 5t^3 - 10t^2\) So the factored polynomial is correct.

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is a fundamental concept when dealing with algebraic expressions. It refers to the highest number and/or variable expression that can divide each term of a polynomial evenly. To find the GCF in a given polynomial, you need to look at both the numerical coefficients and the variable parts of each term. Let's explore the steps involved in determining the GCF.

• Numerical Coefficients: Begin by identifying the greatest number that can divide each numerical coefficient without leaving a remainder. In the example \(-5t^3 + 10t^2\), the coefficients are \-5\ and \10\. The largest number that evenly divides both is \5\. However, since we have a negative in front, we'll take the negative GCF as \-5\.
• Variable Part: Look at the power of each variable present in the polynomial terms. In our example, \t^3\ and \t^2\ are the variable components. The GCF will have the lowest power of the variable, here, \t^2\.
• Combine: Combine both the numerical and variable GCF to be \-5t^2\, which can now be factored out from the polynomial.

Recognizing and factoring out the GCF simplifies polynomials, making them much easier to work with in further algebraic operations.

Polynomial Expressions

Polynomial expressions are made up of terms, which are separated by addition or subtraction signs. A term can be a standalone number (constant), a variable, or a number multiplied by one or more variables raised to a power. Understanding the structure of polynomial expressions is key to mastering algebraic techniques, such as factoring.
Polynomials are categorized based on the number of terms and the powers of the variables they contain:

• Number of Terms: A polynomial can have one term (monomial), two terms (binomial), or more than two terms (polynomial).
• Degree of a Polynomial: The degree is the highest power of the variable in the expression. For example, in \(-5t^3 + 10t^2\), the degree is 3, because \t^3\ is the term with the highest power.
• Standard Form: Commonly, polynomials are written in standard form, which means arranging the terms from the highest to the lowest power of the variable.

Factoring polynomial expressions often involves identifying patterns or employing the GCF to simplify the expression.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. These form the backbone of algebra, allowing us to represent real-world problems in mathematical form. Unlike numerical expressions, algebraic expressions can contain letters that stand for numbers, which we call variables.
Here are basic elements of algebraic expressions:

• Variables: Symbols that represent unknown or variable quantities, often letters like \x\, \y\, or \t\.
• Constants: Fixed values, such as numerical coefficients, that don't change.
• Operations: Mathematical actions such as addition, subtraction, multiplication, and division applied to numbers and variables.

To manipulate algebraic expressions, such as factoring or simplifying, it's crucial to understand how numbers and variables interact under these operations. For example, when factoring the expression \(-5t^3 + 10t^2\), we used the greatest common factor to simplify it to \(-5t^2(-t + 2)\). This simplification process makes solving equations and further algebraic manipulations much more manageable.