Problem 48 - Beginning and Intermediate Algebra

Question

Factor out the greatest common factor. Be sure to check your answer. $$ \text { Factor out }-4 v^{3} \text { from }-4 v^{5}-36 v^{3} $$

Step-by-Step Solution

Answer

The short answer is: \(-4v^5 - 36v^3 = -4v^3(v^2 - 9)\).

Step 1: Find the Greatest Common Factor of Coefficients

First, find the greatest common factor of the coefficients, -4 and -36. The greatest common factor is -4.

Step 2: Determine the Lowest Power of v

Next, examine the expressions and find the lowest power of v. The terms are -4v^5 and -36v^3. The lowest power of v (lowest exponent) is v^3.

Step 3: Factor out the Greatest Common Factor from the Expression

Factor out the greatest common factor we found in steps 1 and 2 (which is -4v^3) from the expression. \(-4v^5 - 36v^3 = -4v^3(v^2 - 9)\)

Step 4: Check your Answer

Multiply the factors back together to ensure the answer is correct. \(-4v^3(v^2 - 9) = -4v^5 - 36v^3\) Since the original expression is equal to the factored one, the factored form is correct.

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is a key concept in simplifying polynomials. It’s the largest number that divides all coefficients in an expression. In terms of both constants and variables, it’s the most significant factor that can be factored out from each term in a polynomial.
To find the GCF of the numbers -4 and -36:

• List the factors of -4: 1, 2, 4, and -1, -2, -4.
• List the factors of -36: 1, 2, 3, 4, 6, 9, 12, 18, 36, and their negative counterparts.
• The greatest factor that appears in both lists is 4, and when considering negative factors, it's -4.

In algebraic terms, the GCF doesn't stop with coefficients. It includes the variables. You also look for the smallest power of a common variable in all terms. Therefore, in expressions like \(-4v^5\) and \(-36v^3\), the GCF related to 'v' is \(v^3\).
Understanding the GCF allows you to simplify expressions and make further calculations manageable.

Polynomial Expressions

Polynomial expressions are algebraic structures comprising variables and constants. They are assembled using operations of addition, subtraction, multiplication, and non-negative integer exponents. A polynomial's degree is determined by its highest exponent.
In the exercise, the polynomial is \(-4v^5 - 36v^3\), indicating two terms:

• -4\(v^5\): First term, with a coefficient of -4, and the variable raised to the fifth power.
• -36\(v^3\): Second term, with a coefficient of -36, and the variable raised to the third power.

The structure of polynomial expressions forms the basis of algebraic manipulation. Recognizing different polynomial forms helps in simplifying and solving higher-degree mathematics problems efficiently.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions into more useful or easily understandable forms. It is crucial for solving equations, simplifying expressions, and performing factorization.
Factorization, which is one of the forms of algebraic manipulation, entails expressing a polynomial as a product of its factors. To factor \(-4v^5 - 36v^3\), we look for common factors among terms:

• Identify the GCF: here it is \(-4v^3\).
• Rewrite and factor out \(-4v^3\): \[-4v^5 - 36v^3 = -4v^3(v^2 - 9)\]

Factoring helps in simplifying expressions, solving polynomial equations, and is fundamental in calculus and higher-level math.

Exponents and Powers

Exponents and powers give a beautiful structure to mathematical notation by indicating repeated multiplication of a number by itself. They are essential in understanding the degree and behavior of polynomials.
In the expression \(-4v^5 -36v^3\), exponents indicate the power to which the variable 'v' is raised:

• The first term has \(v^5\), meaning 'v' is multiplied by itself 5 times.
• The second term has \(v^3\), indicating 'v' is multiplied by itself 3 times.

Understanding exponents is crucial when factoring, as one must determine the lowest power to factor out. Recognizing the role of exponents aids in comprehending the scale and change rates within polynomial and algebraic functions, making them an indispensable part of mathematics.