Problem 33 - College Algebra with Modeling and Visualization

Question

Factor the expression completely. \(x^{2}+7 x+10\)

Step-by-Step Solution

Answer

The expression \(x^2 + 7x + 10\) factors to \((x + 5)(x + 2)\).

Step 1: Identify the expression

The expression we need to factor is a quadratic polynomial: \( x^2 + 7x + 10 \).

Step 2: Understand the Factorization Process

For the quadratic expression \( ax^2 + bx + c \), factorization involves finding two binomials of the form \((x + p)(x + q)\) such that when multiplied, they give us the original polynomial.

Step 3: Find Numbers that Multiply to c and Add to b

Here, \( c = 10 \) and \( b = 7 \). Look for two numbers that multiply to 10 and add up to 7. The numbers 5 and 2 satisfy these conditions since \(5 \times 2 = 10\) and \(5 + 2 = 7\).

Step 4: Write the Factorized Form

Using the numbers found, the factorized form of the expression is \((x + 5)(x + 2)\). You can check by expanding this expression: \((x + 5)(x + 2) = x^2 + 2x + 5x + 10 = x^2 + 7x + 10\), which matches the original polynomial.

Key Concepts

Quadratic Polynomials

Quadratic polynomials, simply put, are expressions of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). These expressions take their name from the term 'quadratic', which is derived from the Latin word "quadratus", meaning square. The leading term \( ax^2 \) provides this square aspect, being a second-degree polynomial.

These types of polynomials are commonly encountered in algebra due to their straightforward form and fundamental role in various mathematical applications, such as solving quadratic equations, graphing parabolas, and modeling natural phenomena. A quadratic polynomial describes a parabolic path when plotted on a graph, thanks to the squared term. This simple yet versatile structure makes it an essential concept in mathematics, which students often explore through tasks like factorization or finding roots.

Understanding how to manipulate and solve quadratic polynomials is a key skill for any math student, as it paves the way for tackling more complex algebraic concepts and real-world problems.

Binomial Expressions

Binomial expressions are algebraic expressions made up of two terms connected by either a plus or minus sign. A typical form of a binomial is \( (x + p) \) or \( (x - p) \). This dual-term structure is significant in factorization since, for quadratic polynomials, binomials often serve as the building blocks for rewriting the expression in its factorized form.

When working with quadratic polynomials like \( x^2 + 7x + 10 \), the goal of factorization is to break down the polynomial into a product of two binomial expressions. For example, in our exercise, the factorized form \( (x + 5)(x + 2) \) consists of binomials.

• These binomials are structured such that their product will expand back into the original quadratic polynomial.
• The combined middle terms of the expanded binomials give the linear coefficient \( b \).
• The constant \( c \) is the product of the constant terms from each binomial.

Mastering how binomial expressions factor into and out of quadratics equips students with a powerful tool for simplifying and solving quadratic equations.

Factorization Process

The factorization process for quadratic polynomials is a systematic method of expressing the polynomial as a product of simpler components, often two binomials. The process hinges on the relationships between the coefficients of the polynomial. Here's how it works, step-by-step:

• Start by identifying the coefficients \( a \), \( b \), and \( c \) in the quadratic expression \( ax^2 + bx + c \).
• Look for two numbers, \( p \) and \( q \), that multiply together to equal \( ac \) (the product of the first and last coefficients) and add up to \( b \) (the middle coefficient).
• Write the quadratic expression as a product of two binomials: \( (x + p)(x + q) \).
• Verify the factorization by expanding the binomials to ensure it equals the original quadratic \( x^2 + bx + c \).

In our example, the expression \( x^2 + 7x + 10 \) was factorized using this process, identifying 5 and 2 as the numbers fulfilling these conditions. After factorizing, the expression becomes \( (x + 5)(x + 2) \), which multiplies back to confirm the original polynomial. Understanding this process helps in decomposing complex polynomials into manageable parts, greatly simplifying algebraic equations and solutions.