Problem 104 - Elementary Algebra

Question

Factor each trinomial of the form \(x^{2}+b x y+c y^{2}\). \(p^{2}-8 p q-65 q^{2}\)

Step-by-Step Solution

Answer

(p - 13q)(p + 5q)

Step 1: Identify coefficients

First identify the coefficients in the trinomial. Here, the trinomial is p^{2} - 8 p q - 65 q^{2}. Identify the coefficients as follows: a = 1, b = -8, c = -65.

Step 2: Find two numbers that multiply and add

Find two numbers that multiply to ac (a * c) and add to b. In this case, a * c = 1 * (-65) = -65 and b = -8. The two numbers are -13 and 5. -13 * 5 = -65 -13 + 5 = -8.

Step 3: Rewrite the middle term

Rewrite the middle term of the trinomial using the two numbers found. p^{2} - 8pq - 65q^{2} = p^{2} - 13pq + 5pq - 65q^{2}

Step 4: Factor by grouping

Group the terms so that each group can be factored: p^{2} - 13pq + 5pq - 65q^{2} = p(p - 13q) + 5q(p - 13q).

Step 5: Factor out the common binomial

Factor out the common binomial: (p - 13q) is common in both terms. (p - 13q)(p + 5q).

Key Concepts

Trinomial

A trinomial is a polynomial with three terms. For example, in the given exercise, the expression is \( p^{2} - 8pq - 65q^{2} \).
To factor a trinomial, we break it into simpler expressions that multiply together.
Here, we're dealing with a trinomial of the form \( x^{2} + bxy + cy^{2} \).

• The first term is squared \((x^{2})\),
• the second term involves the product of two different variables \((bxy)\),
• and the third is another squared term \((cy^{2})\).

Understanding this structure allows us to follow each step of the factoring process effectively.

Factor by Grouping

Factoring by grouping is a method used for factoring polynomials.
It involves grouping terms with common factors and then factoring out the greatest common factor from each group.

In our example, the expression after rewriting the middle term becomes:
\[ p^{2} - 13pq + 5pq - 65q^{2} \].

• We group terms: \( (p^2 - 13pq) + (5pq - 65q^2) \).
• Then, we factor out the common factor from each group: \( p(p - 13q) + 5q(p - 13q) \).

This grouping allows us to factor further because both groups now contain the same binomial factor \((p - 13q)\).
Thus, we rewrite the expression as \((p - 13q)(p + 5q)\).
This final expression is the factored form of the given trinomial.

Algebra

Algebra plays a crucial role in solving problems like trinomial factoring.
It involves working with variables, constants, and arithmetic operations to solve equations and understand relationships between numbers.
Let's break down a few key terms and operations we used in our trinomial factoring task:

• **Coefficients**: These are the numerical factors of the terms. In the trinomial \( p^{2} - 8pq - 65q^{2} \), we identified the coefficients as: \( a = 1 \), \( b = -8 \), and \( c = -65 \).
• **Rewriting the Middle Term**: We needed to find two numbers that multiply to the product of the first and third coefficients and add to the middle coefficient. We rewrote \(-8pq\) as \(-13pq + 5pq\) to make factoring easier.
• **Common Factors**: In algebra, factoring out common factors simplifies expressions. For the grouped terms, we pulled out common factors: \( p \) from \( p^{2} - 13pq \), and \( 5q \) from \( 5pq - 65q^{2} \).

By mastering these algebraic techniques, you can handle a wide range of polynomial expressions and factor them efficiently.
Keep practicing to become more confident and skillful in algebra!