Problem 12 - Algebra 1: Concepts and Skills

Question

Choose the correct factorization. $$ x^{2}+7 x+12 $$ $$ \begin{aligned} &a.\quad(x+6)(x+2)\\\ &b.\quad(x+4)(x+3) \end{aligned} $$

Step-by-Step Solution

Answer

The correct factorization of \(x^{2}+7x+12\) is \((x+4)(x+3)\).

Step 1: Identify the quadratic equation

The given expression that needs to be factorized is \(x^{2}+7x+12\). It is a quadratic equation of the standard form \(ax^{2}+bx+c\), where \(a=1, b=7, c=12\). The quadratic equation generally factorizes into two binomial expressions.

Step 2: Check the factorizations

Now, check the provided options for the potential factorization of the given quadratic equation. This can be done by expanding the provided factorizations and checking if it equals to the given quadratic expression.

Step 3: Expand Option a

First, look at option a: \((x+6)(x+2)\). Expanding this gives: \(x^{2}+8x+12\). While this is a valid quadratic equation, it's not the same as the one that was originally presented to factorize, so option a is incorrect.

Step 4: Expand Option b

Second, look at option b: \((x+4)(x+3)\). Expanding this gives: \(x^{2}+7x+12\). This matches the quadratic equation to be factorized.

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of the second degree. It typically takes the form:

• \( ax^2 + bx + c = 0 \)

where:

• \( a, b, \) and \( c \) are constants,
• \( x \) represents an unknown variable,
• and \( a eq 0 \) because if \( a = 0 \), it turns into a linear equation.

Understanding the components of a quadratic equation is crucial because they determine the behavior and properties of the parabola it represents. The term \( ax^2 \) indicates its degree, which means any graph of a quadratic equation forms a U-shaped curve called a parabola.
Quadratic equations can be solved in several ways such as factorization, using the quadratic formula, or graphing. When factorizing, one looks for two numbers that multiply to give \( ac \) and add to give \( b \). This is the process used when factoring the quadratic given in the exercise.

Binomials

Binomials are algebraic expressions containing two distinct terms. They are often written in the form:

• \( (a + b) \)
• or \( (a - b) \)

where both \( a \) and \( b \) are constants or variables.
A binomial, when multiplied by another binomial, can help in expressing or solving quadratic equations. For example, the expansion of the binomial expression \((x + 4)(x + 3)\) follows the distributive property, where each term in the first binomial multiplies each term in the second. The result \( x^2 + 7x + 12 \) shows that binomials are extremely useful in polynomial manipulation and simplification, aiding in verifying factorization of a quadratic expression.
Understanding binomials is essential as they form the building blocks for more complex algebraic equations and provide a simpler pathway to solve quadratic equations efficiently.

Polynomial Expansion

Polynomial expansion involves multiplying polynomials together and combining like terms. This is especially useful when dealing with quadratic equations which are factored into binomials and then expanded to verify the solution.
When expanding polynomials, every term in the first polynomial must be multiplied by every term in the second one. For example, expanding the binomials \((x + 4)(x + 3)\) involves multiplying each term in the first parenthesis by each term in the second. This results in:

• \(x \times x = x^2\)
• \(x \times 3 = 3x\)
• \(4 \times x = 4x\)
• \(4 \times 3 = 12\)

Combining these gives us \(x^2 + 7x + 12\).
The polynomial expansion not only simplifies solving but also validates the factorization process, ensuring that the original quadratic equation's integrity is maintained. Mastering polynomial expansion creates a robust foundation necessary for tackling more complicated algebraic expressions and functions.