Problem 36 - Beginning and Intermediate Algebra

Question

Factor out the greatest common factor. Be sure to check your answer. $$21 b^{4} d^{3}+15 b^{3} d^{3}-27 b^{2} d^{2}$$

Step-by-Step Solution

Answer

The factored expression with the greatest common factor is: \(\boxed{3b^2d^2(7b^2d + 5bd - 9)}\).

Step 1: Find the GCF of the coefficients

Calculate the GCF for the coefficients 21, 15, and -27. The GCF for these numbers is 3.

Step 2: Find the GCF of the variables

Find the variable with the lowest exponent in each term for both b and d. The variables with the lowest exponents are \(b^2\) and \(d^2\). So, the GCF of the variables is \(b^2 d^2\).

Step 3: Write the original expression as the GCF multiplied by the remaining expression

The GCF is 3 and the GCF for variables is \(b^2 d^2\), so the expression can be written as: \[3b^2d^2(7b^2d + 5bd - 9)\]

Step 4: Check the answer by multiplying the GCF and the remaining expression

Multiply the GCF and the remaining expression to verify the factored form: \[3b^2d^2(7b^2d + 5bd - 9) = 21 b^{4} d^{3}+ 15 b^{3} d^{3} - 27 b^{2} d^{2}\] Since the multiplication results in the original expression, our factored form is correct. The polynomial factored out with the greatest common factor is: \[\boxed{3b^2d^2(7b^2d + 5bd - 9)}\]

Key Concepts

Factoring Polynomials

Factoring polynomials can be a fun puzzle to solve in algebra! It involves breaking down a complex expression into simpler parts that, when multiplied together, give back the original expression. In our exercise, we factor out the greatest common factor (GCF) first. By doing this, we're looking for the largest number and variable combination that divides all parts of the polynomial evenly. Once we've identified the GCF, we divide each term by it to uncover a simpler expression. This process of factoring polynomials not only reveals the structure of the polynomial but can also help in solving equations.

Algebraic Expressions

Algebraic expressions are like little packages of information containing numbers, variables, and operations. Just like unpacking a gift to see what's inside, you analyze an expression by breaking it down into its components. In the exercise, the expression \(21 b^{4} d^{3}+15 b^{3} d^{3}-27 b^{2} d^{2}\) consists of three terms. Each term is made up of coefficients and variables raised to powers. By understanding each part, especially when we factor them, we can manipulate the entire expression for solving, simplifying, or rewriting it in a more convenient form.

Mathematical Problem Solving

Mathematical problem solving is like being a detective—it's all about finding a strategy to solve a case! When approaching a problem, the first step is understanding what's being asked. For factoring the given polynomial, the goal is to find the greatest common factor and use it to simplify the expression. Our strategy is to methodically determine the GCF of both the numbers and variables involved. Then, we verify our solution by checking if the factorized expression can be expanded to regain the original form. With practice, problem-solving becomes an intuitive and rewarding process!

Exponents and Variables

Exponents and variables are both key players in algebra. Variables represent unknowns that can change, and exponents tell us how many times to multiply a variable by itself. In our exercise, noticing patterns in exponents is crucial for factoring. When multiple terms share variables and exponents, the focus is on finding the smallest exponent for each shared variable. This reveals which base form, such as \(b^2 d^2\), can be factored out. Understanding how exponents and variables interact helps unravel complex expressions and makes solving them a systematic adventure.