Problem 33 - Beginning and Intermediate Algebra

Question

Factor completely, if possible. Check your answer. $$24+14 d+d^{2}$$

Step-by-Step Solution

Answer

The factored form of the given expression is \(2(12 + 7d + \frac{1}{2}d^2)\).

Step 1: Identify the common factors

First, let's look for any common factors in the coefficients and the variables. Both the constant term (24) and the linear term (14d) have coefficients of 2, so we have a common factor of 2 that we could factor out from the entire expression.

Step 2: Factor out the common factor

Now that we have identified the common factor (2), let's factor it out from the expression. \(2(12+7d+d^2/2)\)

Step 3: Try factoring further

Next, let's see if we can factor the remaining expression inside the parentheses further: \(12 + 7d + \frac{1}{2}d^2\). Unfortunately, we don't see any other common factors or any form that allows us to factor further, such as a perfect square trinomial or a difference of squares.

Step 4: Write the final factored form

Since we cannot factor the expression any further, our final factored form is: \(2(12 + 7d + \frac{1}{2}d^2)\)

Step 5: Check your answer

Let's make sure our factored form is equivalent to the original expression. We can check this by redistributing the 2 back into the expression inside the parentheses. \(2(12) + 2(7d) + 2(\frac{1}{2}d^2)\) \(24 + 14d + d^2\) Our factored form, \(2(12 + 7d + \frac{1}{2}d^2)\), is equivalent to the original expression, \(24 + 14d + d^2\).

Key Concepts

algebra

Algebra is the branch of mathematics that operates using symbols and variables to solve various equations and problems. In this exercise, we are using algebra to factor a polynomial expression. Factoring in algebra involves breaking down a complex expression into simpler components or "factors." This process can simplify the expression and make solving equations more manageable.

In the given expression, which is a polynomial, we have terms separated by addition. Each term is made up of numbers and the variable "d," with different powers of the variable, which is a hallmark of polynomial expressions. Using algebra, we manipulate the terms and explore which parts of the expression can be factored out evenly, simplifying our task of solving or transforming complex problems.

common factors

To successfully factor a polynomial, it is crucial first to identify any common factors among the terms. A common factor is a number or variable that can evenly divide each term in the expression. This step simplifies the polynomial by reducing it through division.

In the exercise, the expression is composed of three terms: 24, 14d, and \(d^2\). By examining the coefficients (the numerical parts of each term), we notice that 2 is a common factor for the numerical parts of the terms, specifically from 24 and 14. For factoring purposes, identifying such a common factor helps us rewrite the original polynomial in a simplified form, which may reveal a pattern or another factorization option we could pursue. Hence, we factor out the common factor of 2.

factored form

The factored form of a polynomial is an expression that is rewritten as a product of its factors. Achieving the factored form helps in simplifying expressions and solving equations. In this exercise, after identifying that the number 2 can be factored out from all terms, we are left with the expression \(2(12 + 7d + \frac{1}{2}d^2)\).

Factoring often aims to simplify equations, making values or solutions easier to find. When no further factoring is possible, either due to a lack of patterns like a perfect square or a special math property, the expression we end with is our final factored form. In this case, attempts to factor further did not result in a simpler expression, so the current state is the simplest we can achieve given the information.

check your answer

After factoring an expression, it is essential to check your work to ensure accuracy. Checking involves redistributing any factored values back into the expression inside the parentheses to see if it matches the original. This step confirms that no mistakes were made in the factoring process.

From our solution, the factored form is \(2(12 + 7d + \frac{1}{2}d^2)\). By multiplying each term inside the parentheses by 2, we performed the check:

• \(2 \times 12 = 24\)
• \(2 \times 7d = 14d\)
• \(2 \times \frac{1}{2}d^2 = d^2\)

Adding these back together gives us the original expression \(24 + 14d + d^2\). This confirms that our factored form is correct. Checking your answer not only provides assurance of accuracy but also helps in better understanding the relationship between an expression and its factors.