Problem 68 - Elementary and Intermediate Algebra

Question

Factor. If an expression is prime, so indicate. $$ 14 y^{2}+11 y+2 $$

Step-by-Step Solution

Answer

The factored form is \((7y + 2)(2y + 1)\).

Step 1: Identify coefficients

The given quadratic expression is \(14y^2 + 11y + 2\). Identify the coefficients: \(a = 14\), \(b = 11\), and \(c = 2\).

Step 2: Check for a multiplication pattern

We need to find two numbers that multiply to \(a imes c\) which equals \(14 imes 2 = 28\) and add up to \(b = 11\).

Step 3: Determine suitable numbers

The numbers that satisfy these conditions are \(7\) and \(4\) since \(7 imes 4 = 28\) and \(7 + 4 = 11\).

Step 4: Rewrite the middle term

Rewrite \(11y\) as \(7y + 4y\). The expression becomes \(14y^2 + 7y + 4y + 2\).

Step 5: Group terms

Group the terms into two pairs: \((14y^2 + 7y) + (4y + 2)\).

Step 6: Factor each group

Factor out the greatest common factor from each pair. From \(14y^2 + 7y\), factor out \(7y\) to get \(7y(2y + 1)\). From \(4y + 2\), factor out \(2\) to get \(2(2y + 1)\).

Step 7: Combine factored terms

Notice that both terms contain \(2y + 1\). Factor out \(2y + 1\) to get \((7y + 2)(2y + 1)\).

Step 8: Check factors

Expand \((7y + 2)(2y + 1)\) to ensure it equals the original expression. This results in \(14y^2 + 7y + 4y + 2 = 14y^2 + 11y + 2\), confirming the factorization is correct.

Key Concepts

Quadratic Expressions

Understanding quadratic expressions is crucial for many mathematical calculations. A quadratic expression is any polynomial where the highest power of the variable is 2. For example, in the expression \(14y^2 + 11y + 2\), the term \(14y^2\) indicates it is quadratic because of the exponent of 2 on \(y\).
This expression is a trinomial, composed of three terms: the quadratic term \(14y^2\), the linear term \(11y\), and the constant \(2\). The arrangement is always descending in powers of the variable. Quadratic expressions are foundational in algebra and are often considered for various applications such as graphing or solving equations.
Recognizing the structure of a quadratic expression helps in applying various algebraic techniques such as factoring, which simplifies the expression into a product of simpler terms. This is particularly useful when solving quadratic equations, as it transforms the equation into a more manageable form.

Coefficient Identification

When working with quadratic expressions, identifying the coefficients correctly is a vital step. Coefficients are the numbers in front of the variables. In the quadratic expression \(14y^2 + 11y + 2\), you would identify:

• \(a = 14\) for the quadratic term \(y^2\)
• \(b = 11\) for the linear term \(y\)
• \(c = 2\) as the constant term

The coefficients serve as the backbone of the quadratic expression and guide the process of factoring. Knowing \(a\), \(b\), and \(c\) helps you assess relationships between the terms, such as multiplication patterns or sums.
Correct coefficient identification is essential as one small mistake can lead to incorrect factorization, hence errors in solving the overall problem. Being meticulous in assigning these coefficients is a fundamental skill in algebra.

Factorization Techniques

Factorization involves rewriting a quadratic expression as a product of simpler expressions. The goal is to break down \(14y^2 + 11y + 2\) into factors. One popular technique is to find the product \(a \times c\) and seek two numbers that multiply to this product and add to \(b\).

For example, for \(14 \times 2 = 28\), you look for two numbers that multiply to 28 and add to 11. The correct pair is 7 and 4 because \(7 \times 4 = 28\) and \(7 + 4 = 11\).
This information lets you rewrite \(11y\) as \(7y + 4y\). Hence, you can organize and factor the expression in steps:

• Group terms: \((14y^2 + 7y) + (4y + 2)\)
• Factor each group individually: \(7y(2y + 1) + 2(2y + 1)\)
• Combine factored terms: \((7y + 2)(2y + 1)\)

These steps confirm the expression has been factored correctly. Practicing these techniques helps improve problem-solving skills and confidence in handling quadratic expressions.