Problem 102 - Elementary Algebra

Question

Factor each trinomial of the form \(x^{2}+b x y+c y^{2}\). \(p^{2}-16 p q+63 q^{2}\)

Step-by-Step Solution

Answer

(p - 7q)(p - 9q)

Step 1: Identify the form

The given trinomial is in the form of \(x^2 + bxy + cy^2\). For this problem, \(p^2 - 16pq + 63q^2\), identify \(x = p\), \(y = q\), \(b = -16\), and \(c = 63\).

Step 2: Find factors of \(c\)

Identify two numbers that multiply to \(c\) and add to \(b\). We need factors of 63 that add to -16. These factors are -7 and -9 since \(-7 \times -9 = 63\) and \(-7 + -9 = -16\).

Step 3: Write the factored form

Using the factors identified, rewrite the trinomial as the product of two binomials: \( (p - 7q)(p - 9q) \).

Key Concepts

Factoring Trinomials

Factoring trinomials might seem difficult at first, but it becomes much easier with a step-by-step approach. A trinomial is a polynomial with three terms. The trinomial in this exercise is of the form \(x^2 + bxy + cy^2\). Here, identifying the proper values of \(b\) and \(c\) is crucial for proper factoring.
For the given problem, \(p^2 - 16pq + 63q^2\), we can recognize:

• \(b = -16\)
• \(c = 63\)

The goal is to rewrite this trinomial as the product of two binomials. Factors of \(c\) that add up to \(b\) play a key role. Factors for \(63\) that add up to \(-16\) are \(-7\) and \(-9\). Thus, the correct factored form is \((p - 7q)(p - 9q)\).

Quadratic Equations

A quadratic equation is a second-degree polynomial equation of the form \ax^2 + bx + c = 0\. In this exercise, the example can be visualized as a quadratic equation in terms of \(p\) and \(q\).
The general form here is \(p^2 + (-16)pq + 63q^2 = 0\). Quadratic equations can be solved using various methods, such as:

• Factoring
• Completing the square
• The quadratic formula

However, factoring is often the simplest method when it works.
By recognizing the trinomial pattern and identifying suitable factors, we can convert the equation into \((p - 7q)(p - 9q) = 0\). This shows that \(p - 7q = 0\) or \(p - 9q = 0\). Solving these linear equations gives us the possible values for \p\ and \q\.

Polynomial Factorization

Polynomial factorization is breaking down a polynomial into simpler 'factor' polynomials that, when multiplied together, give back the original polynomial. Here, we aimed to factor the quadratic trinomial \(p^2 - 16pq + 63q^2\) into simpler binomials.
The steps involve:

• Recognizing the equation in the standard form \(x^2 + bxy + cy^2\)
• Finding two numbers that multiply to \c\ and add to \b\

In this case, recognizing the factors \(-7\) and \(-9\) for \63\ that add up to \(-16\) is key. The factorization thus yields \((p - 7q)(p - 9q)\). Factorization simplifies solving polynomial equations and is fundamental in algebra.