Problem 35 - Algebra 1: Concepts and Skills

Question

Factor the trinomial. $$ 3 c^{2}-37 c+44 $$

Step-by-Step Solution

Answer

The factored form of the trinomial \(3 c^{2}-37 c+44\) is \((3c-11) (c-4)\)

Step 1: Identify the coefficient of each term

The coefficients of the equation are \(3, -37, 44\) respectively.

Step 2: Find two numbers

Find two numbers that multiply to \(3 x 44\) ..which gives 132, and also the same two numbers would add up to -37 to get the middle term. After some trial and error, those two numbers are found to be -11 and -12.

Step 3: Split the middle term

Split the middle term of the polynomial using the two numbers found in the previous step. This gives us \(3c^{2} -11c -12c +44\).

Step 4: Factor by grouping

Note that the first two terms have a mutual factor of \(c\) and the last two terms have a mutual factor of \(4\). This gives us \(c(3c-11) -4(3c-11)\).

Step 5: Completing the Factoring

The factored form of the equation would be \((3c-11) (c-4)\) since these two factors give the original polynomial expression when multiplied.

Key Concepts

Polynomial Equations

Polynomial equations involve expressions that include variables raised to whole number exponents. In our original exercise, we deal with the trinomial equation \( 3c^2 - 37c + 44 \). Trinomials are a type of polynomial with three terms.
The objective is often to rewrite these equations in their factorable form, if possible, to simplify or solve them.
A polynomial equation can have:

• a constant term which doesn’t contain any variables — here it's 44,
• linear term with the variable raised to the power of one, such as \(-37c\), and
• a quadratic term with the variable raised to the power two, seen in \(3c^2\).

Understanding polynomial equations helps in identifying features and considering appropriate methods like factoring for solving or simplifying.

Factoring by Grouping

Factoring by grouping is a method used to simplify a polynomial into the product of simpler polynomials. It is particularly useful when dealing with trinomials that are not easily factorable by other means. In the step-by-step solution, this method involves breaking down the trinomial equation into four terms and then grouping the terms to find common factors.
Here's what happens:

• First, determine the target factors by multiplying the leading coefficient (3) by the constant (44) to get 132.
• Next, find two numbers that multiply to give 132 and add up to -37; these numbers in our example are -11 and -12.
• Incorporate these numbers into separate expressions and group: \(3c^2 - 11c - 12c + 44\).
• Lastly, factor each group separately: \(c(3c-11) -4(3c-11)\).

The result is the trinomial factored into \((3c-11)(c-4)\).
This method proves advantageous when solving polynomial equations by making it easier to identify solutions.

Quadratic Expressions

Quadratic expressions are a subset of polynomial expressions where the highest exponent of the variable is 2. In the exercise, \( 3c^2 - 37c + 44 \) is the quadratic expression we're working with. Quadratics generally have the form \( ax^2 + bx + c \).
Understanding quadratics:

• They graph as parabolas, displaying symmetrical shapes.
• The solutions to the quadratic equation or the roots can be found through various methods, including factoring, completing the square, or using the quadratic formula.
• In our trinomial, we're factoring to find these components, effectively determining where these roots or solutions occur.

Factoring provides a direct route to these roots by breaking down the expression into products of linear factors. Thus, factoring our trinomial into \((3c-11)(c-4)\) reveals potential solutions or roots directly from the factored components. These methods showcase the utility of understanding and manipulating quadratic expressions effectively.