Problem 102 Factor each trinomial of the for... [FREE SOLUTION]

Chapter 7: Problem 102

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

Short Answer

Expert verified

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

Step by step solution

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 \(c = 63\).

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 \(-7 + -9 = -16\).

Write the factored form

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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91影视

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

Short Answer

Step by step solution

Identify the form

Find factors of \(c\)

Write the factored form

Key Concepts

Factoring Trinomials

Quadratic Equations

Polynomial Factorization

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