Problem 142 - Algebra and Trigonometry

Question

Find all integers b so that the trinomial can be factored. $$x^{2}+b x+15$$

Step-by-Step Solution

Answer

The possible integer values for \(b\) in the trinomial \(x^{2}+b x+15\) that allow for factoring over the integers are -16, 16, -8, 8.

Step 1: Identify the Form of the Trinomial

The given trinomial is in the generic form \(x^{2}+b x+15\). In this form for quadratic expressions, the coefficient of \(x^2\) is '1', \(b\) is the coefficient of \(x\) we want to find, and the constant term is '15'.

Step 2: Recognize the Factoring Pattern

For a quadratic trinomial of form \(x^{2} + b x + c\) to be factorable over the set of integers, \(b\) must be a sum of two integers whose product is equal to \(c\). This means we need to find pairs of integers that multiply to '15' (as this is the value of 'c'), and check their sums (which will give us the possible values for \(b\).

Step 3: Find Pairs of Integers

The pairs of integers that multiply to '15' are ('1','15') and ('-1','-15'), ('3','5') and ('-3','-5'). They correspond to the possible factorizations of 15.

Step 4: Calculate the Sums

The sums for each pair of integers are: 1+15 = 16, -1-15 = -16, 3+5 = 8, -3-5 = -8. These numbers are the possible values for \(b\).

Step 5: State the Result

Therefore, the trinomial \(x^{2}+b x+15\) can be factored over the set of integers when \(b\) equals -16, 16, -8, or 8. Each of these will satisfy the requirement that the coefficient of \(x\) in the quadratic trinomial can be expressed as the sum of two numbers which have a product of 15.

Key Concepts

Quadratic Expression

A quadratic expression is a type of polynomial that is characterized by having a degree of 2. This means the highest power of the variable, usually denoted by \(x\), is squared. These expressions typically take the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are coefficients with \(a eq 0\). Quadratics are fundamental algebraic structures and appear frequently in mathematics, physics, and engineering.

When dealing with a quadratic expression, you're looking at three separate terms:

• \(ax^2\) is the quadratic term.
• \(bx\) is the linear term.
• \(c\) is the constant term.

Factoring is a common method used to simplify these expressions, making them easier to solve, especially when it comes to finding the roots or solutions for the equation \(ax^2 + bx + c = 0\). Once factored, the quadratic can take on a more manageable form of \((x + p)(x + q)\), where \(p\) and \(q\) are the roots or solutions of the equation.

Integer Factorization

Integer factorization is a mathematical process where a number or expression is broken down into its component whole number (integer) factors. In the context of quadratic expressions, it involves finding pairs of numbers whose product equals the constant term \(c\), provided the quadratic is of the form \(x^2 + bx + c\).

Understanding how to perform integer factorization is crucial when you need to factor trinomials. Here's a quick run-through of how it's done:

• Identify the constant term, \(c\), in the quadratic expression.
• Determine all pairs of integers that multiply together to give you \(c\).
• For each pair, calculate the sum of the integers and check if it matches the linear coefficient \(b\).

In our exercise of \(x^2 + bx + 15\), we focused on the product of 15. The pairs \((1, 15), (-1, -15), (3, 5), (-3, -5)\) were considered because each pair multiplies to give 15. Finding one that matches your quadratic ensures it can be expressed as a product of binomials.

Sum and Product of Roots

The concepts of the sum and product of the roots are deeply connected to factoring quadratic expressions. When a quadratic equation \(ax^2 + bx + c = 0\) is factored into \((x + p)(x + q) = 0\), the values \(p\) and \(q\) are known as the roots or solutions.

The sum and product of these roots have specific mathematical relationships with the coefficients of the quadratic expression. According to Vieta's formulas:

• The sum of the roots \(p + q = -b/a\).
• The product of the roots \(pq = c/a\).

These properties provide a valuable method for checking the accuracy of your factored expression. In our exercise \(x^2 + bx + 15\), the task was to ensure that the different pairs of integers that multiply to \(c = 15\) have a sum equalling \(b\), allowing for successful factoring over integers. This understanding of the sum and product of roots simplifies the process and ensures correctness in finding possible values for \(b\).