Problem 60 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ 21 x^{2}+41 x+10 $$

Step-by-Step Solution

Answer

The polynomial \(21x^2 + 41x + 10\) factors to \((3x + 5)(7x + 2)\).

Step 1: Identify a, b, and c

In the quadratic expression \(ax^2 + bx + c\), identify the coefficients \(a = 21\), \(b = 41\), and \(c = 10\). This helps in setting up the factoring process.

Step 2: Find Two Numbers

Find two numbers whose product is \(a \times c = 21 \times 10 = 210\), and whose sum is \(b = 41\). These numbers are 6 and 35, because \(6 \times 35 = 210\) and \(6 + 35 = 41\).

Step 3: Rewrite the Middle Term

Rewrite the middle term \(41x\) as \(6x + 35x\). This results in the expression \(21x^2 + 6x + 35x + 10\).

Step 4: Factor by Grouping

Group the terms into two pairs: \((21x^2 + 6x) + (35x + 10)\). Factor out the greatest common factor from each pair: \(3x(7x + 2) + 5(7x + 2)\).

Step 5: Factor Out the Common Binomial

Notice that \((7x + 2)\) is a common factor. Factor it out: \((3x + 5)(7x + 2)\).

Step 6: Verify the Factorization

Expand \((3x + 5)(7x + 2)\) to confirm it matches the original polynomial. Perform: \(3x imes 7x + 3x imes 2 + 5 imes 7x + 5 imes 2\), simplifying to \(21x^2 + 6x + 35x + 10\), which is \(21x^2 + 41x + 10\). This confirms the factorization is correct.

Key Concepts

Quadratic expressions

A quadratic expression is a type of polynomial where the highest degree of the variable is 2. This means it includes terms up to the power of two. Such expressions follow the standard form: \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(x\) is the variable. Here, \(a\) is the coefficient of \(x^2\), \(b\) is the coefficient of \(x\), and \(c\) is the constant term.
Quadratic expressions are fundamental in algebra because they represent a wide variety of problems, such as projectile motion or area calculations. Recognizing the coefficients accurately is the first step in many problem-solving techniques, including factorization. It's important to note the presence of exactly three terms, unless it has been simplified by grouping or factoring.

Factoring by grouping

Factoring by grouping is a technique used to simplify polynomials by grouping terms that have common factors. This is particularly useful for quadratic expressions where a straightforward factorization isn't immediately obvious.
The method involves separating the middle term in such a way that it allows for common factoring from two groups of terms. In our step-by-step example:

• The middle term \(41x\) is rewritten as \(6x + 35x\) because their sum equals \(41x\) and their product fits the condition \(21 \times 10\).
• This enables the formation of two distinct groups: \((21x^2 + 6x)\) and \((35x + 10)\).
• Each pair is then factored separately, uncovering a common binomial factor, in this case, \((7x + 2)\).

Thus, factoring by grouping transforms a seemingly complex quadratic into a manageable multiplication of binomials.

Greatest common factor

The greatest common factor, or GCF, is an important concept in algebra for simplifying expressions. It represents the largest factor common to two or more numbers or terms. Finding the GCF is pivotal in the process of factoring, as it allows for the cleaning and simplifying of equations.
For instance, during factoring by grouping:

• Identify the GCF in each group. For \((21x^2 + 6x)\), the GCF is \(3x\), because both terms are divisible by \(3x\).
• For \((35x + 10)\), the GCF is \(5\), since both terms share this factor.

By extracting these common factors, each group of terms is reduced to reveal a common binomial, facilitating further simplification. Recognizing and utilizing the GCF is crucial for efficient algebraic manipulation and simplification procedures.

Verification of factorization

Verification of factorization is a crucial step to ensure the accuracy of your factorization process. It is essential to check that the factorized expression, when multiplied out, results in the original expression.
In our example, after factoring to \((3x + 5)(7x + 2)\), it's important to distribute or expand to confirm correctness:

• Multiply each term of the first binomial by each term of the second: \(3x \cdot 7x\), \(3x \cdot 2\), \(5 \cdot 7x\), and \(5 \cdot 2\).
• Simplify these to get \(21x^2 + 6x + 35x + 10\).
• Combine the like terms (\(6x + 35x\)) back into \(41x\), confirming the expression as \(21x^2 + 41x + 10\).

This verification step assures that the factorization is valid and avoids errors in subsequent algebraic calculations.