Problem 44 - Introductory Algebra for College Students

Question

Factor completely. $$3 x^{2}+21 x+36$$

Step-by-Step Solution

Answer

The fully factored form of the given expression \(3x^{2} + 21x + 36\) is \(3(x+3)(x+4)\)

Step 1: Identify the Common Factor

The expression \(3 x^{2} + 21 x + 36\) includes a common factor for all three terms, which is 3. Find the greatest common factor and factor it out using the distributive property.

Step 2: Perform Factoring

Take out the common factor from the trinomial. That gives us: \(3(x^{2} + 7x + 12)\).

Step 3: Factor the Quadratic

Factorize the quadratic expression inside the brackets. Look for two numbers that multiply to 12 and add up to 7. These numbers are 3 and 4. So, we write \(x^{2} + 3x + 4x + 12\) as \((x+3)(x+4)\)

Step 4: Write Down the Complete Factorization

Substitute the factorized quadratic expression back into the equation. The final factorization is \(3(x+3)(x+4)\)

Key Concepts

Greatest Common Factor

When factoring polynomials, a crucial first step is identifying the greatest common factor (GCF). The GCF is the largest number or expression that divides each term in the polynomial. In this case, our polynomial is \(3x^2 + 21x + 36\). To find the GCF, we look for the greatest number that divides 3, 21, and 36 evenly.

• 3 divides into 3 (1 time), 21 (7 times), and 36 (12 times).
• No higher number than 3 does this.

Hence, 3 is the GCF.
Taking out the GCF is like "undoing" the distribution process. It simplifies the polynomial and sets the stage for further factoring. By factoring 3 out of the polynomial, we revise it to \(3(x^2 + 7x + 12)\). This step not only simplifies the equation but also often reveals additional factors of the polynomial.

Quadratic Factoring

Once we have simplified the expression by factoring out the GCF, the next step is to factor the quadratic expression, \(x^2 + 7x + 12\). Quadratic expressions are typically in the form \(ax^2 + bx + c\). To factor them, we need to find two numbers that:

• Multiply to the constant term, here 12.
• Add up to the linear coefficient, which is 7.

In this case, the numbers 3 and 4 work perfectly because:

• 3 * 4 = 12
• 3 + 4 = 7

Armed with these numbers, we can express the quadratic as the product of two binomials: \((x + 3)(x + 4)\). This method of factoring by grouping simplifies the expression into easier parts that multiply to form the original quadratic.

Distributive Property

The distributive property is a helpful tool for both expanding and factoring expressions. It states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. For instance, consider the expression 3 times \((x^2 + 7x + 12)\).
This can be expanded as:

• 3 * \(x^2\) = \(3x^2\)
• 3 * 7x = 21x
• 3 * 12 = 36

This property is what allows us to factor out the GCF in reverse. By taking 3 out, we apply the property to "distribute" the GCF across the polynomial, thereby simplifying it. Understanding the distributive property is essential for both recognizing terms that can be factored and reconstructing a factored expression back to its expanded form. This ensures that the factorization process is both correct and complete.