Problem 44 - Beginning and Intermediate Algebra

Question

Factor by trial and error. $$18 p^{2}+35 p+12$$

Step-by-Step Solution

Answer

The factored expression is \( (6p + 3)(3p + 4) \).

Step 1: Identify the terms of the expression

Let's first identify the terms of the expression: 1. Coefficient of the quadratic term (p^2): 18 2. Coefficient of the linear term (p): 35 3. Constant term: 12 Our given expression is: $$18p^2 + 35p + 12$$

Step 2: Find factors of 18, 35 and 12

Find the factors of the coefficients of the quadratic and linear terms, and the constant: 1. Factors of 18: (1, 18), (2, 9), (3, 6) 2. Factors of 35: (1, 35), (5, 7) 3. Factors of 12: (1, 12), (2, 6), (3, 4)

Step 3: Test possible factor pairs by trial and error

We will try different combinations of factors of 18p^2 and factors of 12 to find the correct pair that creates two binomials to give us the middle term: 35p. Test the pairs using the foil method which stands for First, Outer, Inner, Last: - We can try the combination (2p, 6) and (9p, 2): \((2p + 9)(6p + 2) = 12p^2 + 56p + 18\) Not the correct middle term. - Try the combination (6p, 3) and (3p, 4): \((6p + 3)(3p + 4) = 18p^2 + 33p + 12\) This gives us the correct expression when you expand the two binomials.

Step 4: Write the factored expression

The factored expression is: $$ (6p + 3)(3p + 4) $$

Key Concepts

Trial and Error Method

The trial and error method is like experimenting with different ways to solve a math puzzle. It’s often used in factoring quadratic expressions where we test different sets of numbers to see which combination works best.

In the case of factoring, we are essentially looking for two expressions that multiply together to recreate the original quadratic expression.
Here's how it works:

• Identify and list all possible factors of the quadratic term and constant term.
• Pair these factors in various combinations.
• Check which pair, when multiplied, results in the middle term of the quadratic expression through combination and re-arrangement.

It's crucial because it provides insight into the structure of the quadratic expression, allowing us to find an exact solution through persistent testing.

FOIL Method

The FOIL method is an acronym that stands for First, Outer, Inner, Last. It is a technique used to multiply two binomials and is especially useful in checking your work when you factor quadratic equations by trial and error.

To use FOIL:

• First: Multiply the first terms of each binomial.
• Outer: Multiply the outer terms of the binomials.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms of each binomial.

You then sum all these products to get the expanded form of the binomial product. This method is extremely valuable in confirming that your factors combine correctly to give the original expression when expanded.

Quadratic Terms

Quadratic terms form the core of quadratic expressions. These are terms where the variable is squared (e.g., the term \(p^2\) in \(18p^2 + 35p + 12\)).

There are generally three parts to a quadratic expression:

• Quadratic term: Includes the squared variable, indicating its degree is two, like \(18p^2\).
• Linear term: The term with a non-squared variable, such as \(35p\).
• Constant term: A standalone number, which here is 12.

Understanding these components allows us to effectively apply methods like factoring, which heavily relies on dissecting these terms to simplify or restructure them.

Binomial Expansion

Binomial expansion involves expanding multiplied expressions of the form \((a + b)(c + d)\) using methods like FOIL. It helps in understanding and manipulating quadratic expressions.

Here’s how binomial expansion unfolds:

• When you multiply two binomials, each term in the first binomial multiplies with each term in the second binomial.
• The result is an expression typically in the form of a quadratic, such as \(ax^2 + bx + c\).
• Applying binomial expansion allows us to break down these expressions into understandable components or reverse the process—like in factoring quadratics.

Mastering binomial expansion provides a solid foundation for more complex algebraic tasks and enhances problem-solving skills in algebra.