Problem 54 - College Algebra with Modeling and Visualization

Question

Factor the expression completely. \(7 x^{2}+35 x+42\)

Step-by-Step Solution

Answer

The factored expression is \(7(x + 2)(x + 3)\).

Step 1: Identify the Greatest Common Factor (GCF)

Look at the coefficients of the terms in the expression: 7, 35, and 42. The greatest common factor of these numbers is 7. So, factor out 7 from each term.

Step 2: Factor Out the GCF

Factor out the greatest common factor (7) from the expression: \[7(x^2 + 5x + 6)\]

Step 3: Factor the Quadratic Expression Inside the Parenthesis

Now, focus on the quadratic expression \(x^2 + 5x + 6\). Look for two numbers that multiply to 6 (the constant term) and add up to 5 (the coefficient of \(x\)). These numbers are 2 and 3.

Step 4: Decompose the Middle Term and Factor by Grouping

Rewrite the expression \(x^2 + 5x + 6\) as \(x^2 + 2x + 3x + 6\). Group the terms into two pairs: \[(x^2 + 2x) + (3x + 6)\]Factor out the GCF from each pair: \[x(x + 2) + 3(x + 2)\]

Step 5: Factor the Common Binomial

Notice that \(x + 2\) is a common factor in both terms. Factor \(x + 2\) out:\[(x + 2)(x + 3)\]

Step 6: Complete the Final Factored Form

Insert the factor of 7 we initially factored out back into the expression:\[7(x + 2)(x + 3)\]

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is a vital concept when it comes to simplifying or factoring expressions. It refers to the largest number that evenly divides all the terms in an expression. In our exercise, we deal with the expression \(7x^2 + 35x + 42\). The first task is to find the GCF of the coefficients 7, 35, and 42.

To identify the GCF, we need to determine the prime factorization of each coefficient:

• 7 is a prime number, so its factors are only 1 and 7.
• 35 can be expressed as \(5 \times 7\).
• 42 breaks down to \(6 \times 7\), which is the same as \(2 \times 3 \times 7\).

The GCF is the highest common factor found in the factorizations of each term. Here, the number 7 appears in all three, making it the GCF. Factor it out from the expression, simplifying the process of further factorizations.

Quadratic Expression

A quadratic expression is a polynomial expression of degree two, typically written in the form \(ax^2 + bx + c\). Our simplified expression after extracting the GCF is \(x^2 + 5x + 6\), which is a classic quadratic expression.

To factor this quadratic, you need to find two numbers that both:

• Multiply to give the constant term, which in this case is 6.
• Add up to provide the linear coefficient, which is 5 here.

For this expression, the numbers 2 and 3 fit perfectly, as \(2 \cdot 3 = 6\) and \(2 + 3 = 5\). This pair guides us in rewriting and factoring the quadratic expression. The clearer understanding of the behavior and structure of quadratic expressions allows us to break them into products of linear factors.

Factoring by Grouping

Factoring by grouping is a clever method used when an expression has four terms, as happens when decomposing a quadratic expression for factoring. The expression \(x^2 + 5x + 6\) is rewritten by splitting the middle term, using the numbers identified previously (2 and 3). This gives us \(x^2 + 2x + 3x + 6\).

We then group these terms into pairs:

• \(x^2 + 2x\)
• \(3x + 6\)

Now, factor out the GCF from each pair:

• From \(x^2 + 2x\), factor out \(x\) to get \(x(x + 2)\).
• From \(3x + 6\), factor out 3 to get \(3(x + 2)\).

Notice both terms contain the common factor \(x + 2\). We can factor this out and end up with the product \((x + 2)(x + 3)\). This technique is particularly useful and intuitive for handling complex expressions, making them simpler to deal with.