Problem 117 - College Algebra Essentials

Question

In each exercise, replace the boxed question mark with an integer that results in the given product. Some trial and error may be necessary. $$(4 x+1)(2 x-?)=8 x^{2}-10 x-3$$

Step-by-Step Solution

Answer

The value of '?' that makes the given product true is -3.

Step 1: Identify the unknown

In the given expression, we are tasked with finding the value of the unknown '?' which is involved in the multiplication of binomial expressions. This unknown integer affects the resulting trinomial expression after applying distributive property.

Step 2: Evaluate the trinomial

The given trinomial is \(8x^{2} - 10x - 3\). Here, \(8x^{2}\) is the first term, \(-10x\) is the second term and \(-3\). Compare these terms with the coefficient of the resulting trinomial under the distributive property.

Step 3: Apply the distributive property

Apply the distributive property (FOIL) to \((4x + 1)(2x - ?)\). On multiplying the binomials, the first term of trinomial \(8x^{2}\) comes from multiplying the first terms of binomials, \(4x * 2x\). The second term of the trinomial \(-10x\) comes from adding the product of the outer and inner terms of binomials, \((4x * -?) + (1 * 2x)\). The third term of trinomial \(-3\) comes from multiplying the last terms of binomials, \(1 * -?)\).

Step 4: Form equations and solve for the unknown

Now, from the second and third terms of trinomial compare with FOIL products, we can get two equations in terms of '?'. They are \[4? + 2 = -10\] and \[-? = -3\]. By solving these two equations, we can find out the value of '?'.

Step 5: Verify the solution

After getting the value of '?', replace '?' in the original expression and verify by equating to given trinomial expression. If the trinomial expression equates, then '?' value is correct.

Key Concepts

Solving Equations

When we solve equations, our goal is to find the unknown variable that satisfies the expression. In the exercise, we're looking to find the value of '?' that, when used in the equation \((4x+1)(2x-?) = 8x^{2}-10x-3\), results in a true statement.
To solve, we often rearrange and simplify the equation to isolate the unknown. For this problem, we can form two smaller equations derived from comparing each part of the expression with the known trinomial.

• The first equation is derived from comparing the trinomial's middle term: \(4? + 2 = -10\)
• The second equation from the constant term: \(-? = -3\)

By solving these equations, we find '?' step by step, allowing us to verify if our solution fits all parts of the original equation.

Distributive Property

The distributive property is a fundamental concept in algebra that involves multiplying a single term by terms inside a bracket. It helps in expanding expressions and simplifying them in the process.
In the context of this exercise, use the distributive property (often referred to as FOIL for binomials) to expand the product \((4x + 1)(2x - ?)\) by following these steps:

• First: Multiply the first terms: \(4x \times 2x\)
• Outer: Multiply the outer terms: \(4x \times -?\)
• Inner: Multiply the inner terms: \(1 \times 2x\)
• Last: Multiply the last terms: \(1 \times -?\)

This method allows us to break down the multiplication into manageable parts. Then, by matching these products to the terms in the trinomial \(8x^{2}-10x-3\), we can set up equations to solve for the unknown '?'.

Binomials

A binomial is simply an algebraic expression containing two terms, typically involving variables and constants, like \((4x+1)\) or \((2x-?)\). Understanding binomials is key when solving equations involving them, especially when they are multiplied together as in this exercise.
Multiplying binomials requires us to remember the distributive property so we can correctly use the FOIL method. This procedure ensures every term in each binomial is multiplied together.

• When multiplying binomials, start by identifying each term in the binomials.
• Ensure correct application of multiplication by observing which terms are 'first', 'outer', 'inner', and 'last'.

The challenge often lies in managing the negative signs and constant values correctly to match the expanded expression with a trinomial. Identifying the contributions each term in the binomials makes to the final product is crucial. Solving the exercise gives students practice with manipulating binomials, an essential skill for more advanced algebra.