Problem 24 - A Graphical Approach to College Algebra

Question

Factor each trinomial completely. $$6 k^{2}+5 k p-6 p^{2}$$

Step-by-Step Solution

Answer

The expression factors to \((2k + 3p)(3k - 2p)\).

Step 1: Identify the Trinomial Structure

The given expression is a trinomial: \(6k^2 + 5kp - 6p^2\). This polynomial is in the form \(ax^2 + bx + c\), where \(a = 6\), \(b = 5\), and \(c = -6\).

Step 2: Multiply "a" and "c"

Multiply the coefficients of the first term \((a)\) and the last term \((c)\): \(6 \times -6 = -36\).

Step 3: Find Factor Pairs of "ac"

We need to find two numbers that multiply to \(-36\) and add up to the middle coefficient \(b = 5\). The factor pairs of \(-36\) that add up to \(5\) are \(-4\) and \(9\).

Step 4: Split the Middle Term Using Identified Factors

Rewrite the expression, splitting the middle term \(5kp\) into two terms using the factors \(-4\) and \(9\): \[6k^2 - 4kp + 9kp - 6p^2\].

Step 5: Factor by Grouping

Group the terms:- First group: \(6k^2 - 4kp\)- Second group: \(9kp - 6p^2\)Factor each group:- \(6k^2 - 4kp = 2k(3k - 2p)\)- \(9kp - 6p^2 = 3p(3k - 2p)\).

Step 6: Combine the Factored Groups

Notice that \((3k - 2p)\) is a common factor. Thus, the complete factorization is:\[2k(3k - 2p) + 3p(3k - 2p) = (2k + 3p)(3k - 2p)\].

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and mathematical operations such as addition, subtraction, multiplication, and division. These expressions can form simple or complex equations and inequalities. In the context of factorization, understanding these expressions helps in breaking down complex expressions into simpler components. When working with algebraic expressions, it is crucial to identify key elements such as coefficients, variables, and constants. Coefficients are numbers multiplying the variables, while constants are standalone numbers.

Consider the trinomial expression: \(6k^2 + 5kp - 6p^2\). Here, the variables are \(k\) and \(p\), and the coefficients are 6, 5, and -6 respectively. These components will play a vital role when we move on to factoring techniques. By recognizing these parts, we lay the groundwork for more targeted operations crucial for solving algebraic expressions through various mathematical methods, such as factoring.

Polynomials

Polynomials are specific types of algebraic expressions that involve terms in the form of \(ax^n\), where \(a\) is a coefficient and \(n\) is a non-negative integer. They can have one or more terms and are usually classified based on the number of terms they contain: monomials (one term), binomials (two terms), and trinomials (three terms). Understanding polynomials is key to mastering algebraic manipulation and simplification.

In our example trinomial \(6k^2 + 5kp - 6p^2\), there are three terms, each a polynomial term on its own. The highest degree term \(6k^2\) establishes it as a quadratic trinomial, due to the variable \(k\) being squared. Recognizing the structure of a given polynomial helps in selecting the appropriate factoring technique, revealing its simpler polynomial factors.

• Coefficients: Correct identification helps in entire factorization.
• Degrees: Determines the overall degree of the polynomial, guiding the solving strategy.
• Variables: Understanding their roles aids in organizing the expression for further manipulation.

Factoring Techniques

Factoring techniques are strategies used to break down algebraic expressions into simpler, multiplied parts (factors) that, when multiplied together, return the original expression. This process is essential in solving polynomial equations and simplifying expressions. There are several factoring techniques available, each tailored to different types of polynomials. Common methods include common factor extraction, grouping, and special factorizations such as difference of squares or perfect square trinomials.

For the expression \(6k^2 + 5kp - 6p^2\), the method applied is 'factoring by grouping'. This involves rearranging and grouping terms in a way that allows common factors to emerge:

• Identify the middle term: Here, \(5kp\) is split using factor pairs \(-4 \) and \(9\) which multiply to \(-36\) (product of the outer coefficients \(6\) and \(-6\)).
• Group the terms: Rearrange the expression as \(6k^2 - 4kp + 9kp - 6p^2\).
• Factor each group separately: This leads to \(2k(3k - 2p)\) and \(3p(3k - 2p)\), both sharing a common expression \((3k - 2p)\).
• Combine the groups: Achieve the final factorized form \((2k + 3p)(3k - 2p)\).

Mastering these techniques aids greatly in simplifying complex expressions into manageable and solvable equations, allowing for both deeper analysis and easier computation.