Problem 45 - Algebra 1: Concepts and Skills

Question

COMMON FACTOR Factor the expression. $$ 3 k^{2}-39 k+90 $$

Step-by-Step Solution

Answer

The factorized form of \(3k^2 - 39k + 90\) is \((3k - 18)(k - 5)\).

Step 1: Identify the coefficients and the constant

In the trinomial \(3k^2 - 39k + 90\), the coefficient of \(k^2\) is 3, the coefficient of k is -39, and the constant term is 90.

Step 2: Apply the factorization rule

To factorize the trinomial, look for two numbers that multiply to give \(ac\) (product of the coefficient of \(k^2\) and the constant term), and add up to give \(b\) (the coefficient of \(k\)). Here, \(a=3, b=-39, c=90\), and therefore, \(ac=270\). The pair of numbers that meet this condition are -15 and -18 because \(-15 * -18 = 270\) and \(-15 + (-18) = -33\).

Step 3: Rewrite the trinomial

Rewrite the trinomial \(3k^2 - 39k + 90\) as \(3k^2 - 15k - 24k + 90\).

Step 4: Group and factor out

Group the terms and factor by grouping: \(3k(k - 5) - 18(5 - k)\). You can rearrange the second part to make it similar to the first: \(3k(k - 5) - 18(k - 5)\).

Step 5: Final factoring

Now, factoring out \(k - 5\) will give the final factored form of the trinomial: \( (3k - 18)(k - 5)\) .

Key Concepts

Trinomial

A trinomial is a polynomial expression composed of three distinct terms. In most cases, these terms can include variables raised to various powers accompanied by coefficients and a constant term. The general representation of a trinomial looks something like this: \( ax^2 + bx + c \), where each letter represents a specific component of the expression.

• "\(a\)" is the coefficient of the squared term, often setting the leading term in the expression.
• "\(b\)" is the coefficient of the linear term, directly multiplied by the variable itself.
• "\(c\)" is the constant term, standing alone without any variable attached.

Trinomials like \(3k^2 - 39k + 90\) require knowledge of these components to factor and simplify correctly.
Understanding the structure of a trinomial allows us to apply various mathematical strategies, such as factoring, effectively to simplify or solve them.

Coefficients

Coefficients play a critical role in polynomial expressions as they determine the weight of each term. In the context of our example, \(3k^2 - 39k + 90\), we can identify and define the coefficients that affect each term of the trinomial.

• The coefficient for the \(k^2\) term is \(3\), which influences the squared variable.
• The coefficient for the \(k\) term is \(-39\), a negative number affecting the linear component and its variable.
• While 90 is technically the constant term and independent of associated variables, understanding it as part of the whole is crucial for factoring.

Identifying and understanding coefficients is an important first step when factoring trinomials, since they influence the way we approach rewriting and solving the expression.

Factoring by Grouping

Factoring by grouping is a useful method for simplifying trinomials and other polynomial expressions. This approach involves breaking down the expression into parts that can be easily factored separately and then combined to find a common factor.
In the exercise \(3k^2 - 39k + 90\), the method of factoring by grouping is applied as follows:

• Begin by rewriting the middle term in such a way that it can be separated into two terms. In this instance, \(-39k\) is rewritten as \(-15k - 24k\).
• Next, group and factor each pair: \(3k(k - 5)\) and \(-18(k - 5)\).
• Find the common factor from these pairs, realizing that \((k - 5)\) appears in both, which can be factored out.

This results in the successful factorization of the original expression into \((3k - 18)(k - 5)\).
This method is powerful for efficiently handling complex expressions, allowing for a simplified and understandable form, crucial for solving polynomial equations.