Problem 36 - Elementary Algebra

Question

For the following problems, factor the polynomials. $$ -4 x^{2}-12 x-8 $$

Step-by-Step Solution

Answer

Answer: \(-4(x+2)(x+1)\)

Step 1: Identify the Greatest Common Factor (GCF)

We should first check if there is a common factor for all terms in the quadratic polynomial. In this case, the terms are: 1. \(-4 x^{2}\) 2. \(-12 x\) 3. \(-8\) The GCF of these terms is -4. So, we can factor out -4 from the expression.

Step 2: Factor out the GCF

We will factor out -4 from the expression and rewrite the polynomial as a product of the GCF and the remaining terms: $$ -4 x^{2}-12 x-8 = -4(x^{2}+3x+2) $$

Step 3: Factor the remaining quadratic expression

Now, we need to factor the remaining quadratic expression \((x^{2}+3x+2)\). In order to do this, we will look for two numbers that multiply to 2 and add to 3. The numbers 2 and 1 meet these conditions, so we will rewrite the quadratic expression as a product of two binomials: $$ x^{2}+3x+2 = (x+2)(x+1) $$

Step 4: Combine the factored terms

Now we have the GCF and the factored quadratic expression. We will combine these terms to form the final factored expression: $$ -4 x^{2} - 12 x - 8 = -4(x+2)(x+1) $$ The given quadratic polynomial has been factored as \(-4(x+2)(x+1)\).

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is an extremely helpful tool when working with polynomials. When factoring a polynomial, the first step is often to identify its GCF.
This factor is the highest term that divides evenly into all terms of the polynomial. Finding the GCF can greatly simplify the expression, making other factoring steps easier.

For example, consider the polynomial

• \(-4x^2 - 12x - 8\).

Here, each term shares a common factor of -4. Thus,

• The GCF is -4.

By factoring out the GCF, we can rewrite the expression as:

• \(-4(x^2 + 3x + 2)\).

Once the GCF is factored out, it simplifies the polynomial, making the next steps clearer. Remember, finding the GCF is your first strategic move in polynomial factoring!

Quadratic Expression

A quadratic expression is a polynomial of degree two. It generally takes the form

• \(ax^2 + bx + c\),

where \(a\), \(b\), and \(c\) are constants. Understanding the structure of a quadratic expression is crucial because it helps identify strategies for factoring.

For instance, with the expression

• \(x^2 + 3x + 2\),

we can judge that it's a quadratic because the highest power of \(x\) is 2. To factor this, look for two numbers that multiply to give the constant term (2) and add up to the middle coefficient term (3). In this case,

• 2 and 1 are such numbers.

Thus, the quadratic expression can be expressed using two binomials:

• \((x+2)(x+1)\).

Recognizing a quadratic form and knowing how to manipulate it makes the process of finding its factors straightforward and manageable.

Binomials

Binomials are algebraic expressions containing two terms, usually combined by a plus or minus sign. When factoring quadratics, the goal is often to express the quadratic as a product of binomials.
In our given example, once the quadratic expression \((x^2 + 3x + 2)\) is decomposed, it can be rewritten as

• \((x+2)(x+1)\).

Each term in these binomials represents a root or solution to the quadratic equation set to zero.
Being proficient with binomials is key when solving and factoring polynomials, as complex polynomials often break down into smaller, easier-to-handle binomial factors.

With practice, identifying binomials can become intuitive, providing a powerful shortcut in many algebraic problems.