Problem 106 - Elementary Algebra

Question

Factor each trinomial of the form \(x^{2}+b x y+c y^{2}\). \(p^{2}-2 p q-35 q^{2}\)

Step-by-Step Solution

Answer

The factors are \((p - 7q)(p + 5q)\).

Step 1 - Identify coefficients

Observe the trinomial in the form of \(p^{2} + b p q + c q^{2}\). In this case, \(b = -2\) and \(c = -35\).

Step 2 - Find two numbers that multiply and sum correctly

We need to find two numbers that multiply to \( -35 \) (the constant term, \(c\)) and add to \(-2\) (the coefficient of the middle term, \(b\)). These numbers are \(-7\) and \(5\), since \((-7) \times 5 = -35\) and \((-7) + 5 = -2\).

Step 3 - Rewrite the trinomial

Rewrite \(p^{2} - 2 p q - 35 q^{2}\) as \(p^{2} - 7 p q + 5 p q - 35 q^{2}\).

Step 4 - Factor by grouping

Group the expression into two pairs and factor each pair: \((p^2 - 7pq) + (5pq - 35q^2)\). From each pair, factor out the common term: \(p(p - 7q) + 5q(p - 7q)\).

Step 5 - Factor out the common binomial

Factor out the common binomial factor \((p - 7q)\): \((p - 7q)(p + 5q)\).

Key Concepts

algebraic factoring

Algebraic factoring is a fundamental concept in algebra that simplifies complex expressions. Factoring means breaking down a large expression into simpler parts (factors) that, when multiplied together, give the original expression.
For instance, factoring transforms sums or differences of terms into a product of simpler factors.
This is extremely useful in solving equations, simplifying expressions, and even in other areas of mathematics such as calculus.
In the given exercise, we are specifically dealing with the factoring of trinomials.

trinomial expressions

Trinomial expressions are polynomials composed of three terms.
A common class of trinomials has a form like this: ewline + +
For example, consider the trinomial given in the exercise: ewline.
This can be considered as having the structure , where each term plays a specific role in the factoring process.
The quadratic term is p^2 unlike ax^2 in the common form.
The linear term in this case is -2pq, and the constant term is -35q^2.
Understanding how to identify these components is crucial to successfully factoring the trinomial.

polynomial factoring

Polynomial factoring is the process of expressing a polynomial as a product of its factors.
When you factor a polynomial, you break it down into simpler polynomials that can be multiplied together to get the original polynomial.
This is very helpful in solving polynomial equations, simplifying expressions, and understanding the properties of polynomials.
In the exercise, we factored the trinomial into two binomials:

and

.
The process involves several key steps:

• Identifying the coefficients and constants in the trinomial.
• Finding two numbers that multiply to the constant term and add to the coefficient of the middle term.
• Rewriting the trinomial using these numbers.
• Grouping and factoring the resulting pairs of terms.

Each of these steps is essential for correctly factoring the polynomial and arriving at the correct solution.