Problem 30 - Algebra A Combined Function

Question

Factor each trinomial completely. Some of these trinomials contain a greatest common factor (other than 1). Don't forget to factor out the GCF first. $$ x^{2}+19 x+60 $$

Step-by-Step Solution

Answer

The trinomial \(x^2 + 19x + 60\) factors to \((x + 4)(x + 15)\).

Step 1: Identify the coefficients

The given trinomial is \(x^2 + 19x + 60\). Identify the coefficients: \(a = 1\), \(b = 19\), and \(c = 60\).

Step 2: Find two numbers that multiply to c and add to b

We need two numbers that multiply to \(60\) and add to \(19\). These numbers are \(4\) and \(15\) because \(4 \times 15 = 60\) and \(4 + 15 = 19\).

Step 3: Write the trinomial as a product of binomials

We use the numbers found in Step 2 to write the trinomial as a product of two binomials: \((x + 4)(x + 15)\).

Step 4: Check the factorization

Perform the multiplication \((x + 4)(x + 15)\) to verify the factorization. Distribute to find \(x^2 + 15x + 4x + 60\), which simplifies to \(x^2 + 19x + 60\), confirming the factorization is correct.

Key Concepts

greatest common factor

Factoring a trinomial often starts with identifying the greatest common factor (GCF) among its terms. The GCF is the largest number or expression that divides each term evenly.
To find the GCF, look for:

• Shared numerical factors.
• Common variable factors, with the lowest power of shared variables.

In the example, the trinomial is \(x^2 + 19x + 60\). First, check each term to see if there's a number other than 1 that can divide them evenly. Since there isn’t, the GCF is 1, so no need to factor it out here. However, always check for a GCF; it makes factoring easier by simplifying the trinomial first.

trinomial coefficients

A trinomial in the form \(ax^2 + bx + c\) involves three coefficients: \(a\), \(b\), and \(c\). These coefficients determine how the trinomial can be factored.
In the given trinomial \(x^2 + 19x + 60\), we have:

• \(a = 1\): This coefficient is for \(x^2\), making it a simple trinomial.
• \(b = 19\): The coefficient on \(x\) influences how terms combine when searching for factors.
• \(c = 60\): The constant term, crucial for identifying factor pairs.

Understanding these coefficients helps us determine factor pairs that multiply to \(c\) and add to \(b\). Here, finding numbers that multiply to 60 and add up to 19 led us to 4 and 15.

binomial product

Once we identify the correct pair of numbers that combine and multiply to yield the middle and constant terms, the trinomial can be expressed as the product of two binomials.
For \(x^2 + 19x + 60\), we identified the numbers 4 and 15. This lets us rewrite the trinomial as:

\((x + 4)(x + 15)\)

This is known as expressing the trinomial in terms of its binomial product. Each part of the product, \((x + 4)\) and \((x + 15)\), represents one binomial, and together they multiply to create the original trinomial. This form is easier to manipulate and solve, particularly in equations or when looking for roots.

multiplication verification

After factoring a trinomial into binomials, it's crucial to verify the factorization by multiplying the binomials back together. This ensures that the factorization was done correctly.
To verify \((x + 4)(x + 15)\), we perform the multiplication step-by-step:

• First, multiply \(x\) by everything in the second binomial: \(x(x + 15) = x^2 + 15x\).
• Next, multiply 4 by everything in the second binomial: \(4(x + 15) = 4x + 60\).
• Add these results together: \(x^2 + 15x + 4x + 60\).
• Simplify by combining like terms: \(x^2 + 19x + 60\).

This confirms that the original expression is correctly factored, providing accuracy in mathematical solutions.