Problem 115 - College Algebra Essentials

Question

Exercises 115–117 will help you prepare for the material covered in the next section. In each exercise, replace the boxed question mark with an integer that results in the given product. Some trial and error may be necessary. $$(x+3)(x+?)=x^{2}+7 x+12$$

Step-by-Step Solution

Answer

The integer that replaces the question mark is 4.

Step 1: Expansion of Left Side of Equation

Expand the left-hand side of the equation \((x+3)(x+?)\). This results in \(x^2 + 3x + ?x + 3?\).

Step 2: Simplification

Combine the like terms to simplify the expression to \(x^2 + (3+?)x + 3?\)

Step 3: Compare to Right Side of Equation

The expanded and simplified form of the left-hand side now is identical in form to the right-hand side, i.e., \(x^2 + 7x + 12\). The coefficient on \(x\) and the constant term should also match.

Step 4: Equate and Solve

By matching the coefficients and constant terms on either side of the equation, we get 2 equations: \(3+? = 7\) and \(3? = 12\). Solving these equations, the integer that replaces the question mark '?' is found to be 4.

Key Concepts

Polynomial Expansion

Polynomial expansion is a crucial technique in algebra that involves distributing each term of the first polynomial to every term of the second polynomial. For example, when multiplying two binomials, say \(x + a\) and \(x + b\), we use the distributive property to expand the expression, resulting in \(x^2 + ax + bx + ab\).

Let's apply this to the provided exercise. We have the expression \(x+3\) multiplied by \(x+?\), where \( ? \) represents the unknown integer we need to find. The expansion will look like this: \(x \cdot x + x \cdot ? + 3 \cdot x + 3 \cdot ?\). Simplifying this, we get \(x^2 + 3x + ?x + 3?\). As you see, polynomial expansion is essentially about applying the distributive property systematically to every term of the polynomials being multiplied.

Simplification of Expressions

Simplification of expressions is a process used to make expressions easier to interpret or solve. Simplifying generally involves combining like terms, which are terms that have the same variables raised to the same power, and performing arithmetic operations to condense the expression to its simplest form.

In the context of our exercise, after expanding the polynomial, we are left with \(x^2 + 3x + ?x + 3?\). To simplify this, we combine the like terms; in this case, the terms involving \(x\). This results in \(x^2 + (3+?)x + 3?\), where \(3x\) and \(?x\) are combined because they are both terms with \(x\). It's by simplifying expressions that we're able to match and compare coefficients effectively, which brings us to our next concept.

Coefficient Comparison

Coefficient comparison is a method of equating the coefficients (numerical factors) of corresponding terms in two algebraic expressions. This is extremely useful when we have an equation with an unknown variable or number that we need to solve for.

In our example, once we've simplified the expression to \(x^2 + (3+?)x + 3?\), we compare it to the given product \(x^2 + 7x + 12\). To find the value of the unknown \( ? \), we set the coefficients of the corresponding terms equal to each other: the coefficient of \(x\) is \(3+?\) on the left side and 7 on the right side, and the constant terms are \(3?\) and 12, respectively. Solving \(3+? = 7\) gives us \( ? = 4\), and checking \(3 \cdot 4 = 12\) confirms that our solution is consistent. This process illustrates the power of coefficient comparison in solving equations.