Question

Factor each polynomial. $$ 2 x^{2}+3 x+1 $$

Short answer

The factors are \( (2x + 1)(x + 1) \).

Step-by-step solution

Identify the Coefficients

Identify the coefficients of the quadratic polynomial. For the polynomial \( 2x^2 + 3x + 1 \), the coefficients are: \( a = 2 \) \( b = 3 \) \( c = 1 \)

Multiply 'a' and 'c'

Multiply the coefficients of the first term and the constant term: \( a \times c = 2 \times 1 = 2 \)

Find Two Numbers that Multiply to 'ac' and Add to 'b'

Find two numbers that multiply to 2 (the result from Step 2) and add to 3 (the coefficient of the middle term): The numbers 1 and 2 satisfy this requirement because \( 1 \times 2 = 2 \) \( 1 + 2 = 3 \)

Rewrite the Middle Term

Rewrite the polynomial by splitting the middle term using the two numbers found in Step 3: \( 2x^2 + x + 2x + 1 \)

Factor by Grouping

Group the terms in pairs and factor out the greatest common factor (GCF) from each pair: \( x(2x + 1) + 1(2x + 1) \)

Factor Out the Common Binomial

Factor out the common binomial factor: \( (2x + 1)(x + 1) \)

Key concepts

Quadratic Equations

Quadratic equations are a type of polynomial equation that have a degree of 2. This means the highest exponent of the variable (usually x) is 2. The standard form of a quadratic equation is:
\[ ax^2 + bx + c = 0 \]
Here, 'a', 'b', and 'c' are constants (real numbers), and 'x' is the variable we need to solve for. Quadratics can be solved through different methods like factoring, using the quadratic formula, or by completing the square. Factoring is a common method especially when the quadratic can be easily decomposed into binomial factors.
In this exercise, we aim to factor the given quadratic equation 2x² + 3x + 1.

Factoring Techniques

Factoring involves expressing a polynomial as a product of its factors. Several techniques can be employed to factor polynomials. These include:

• **Common Factor Extraction**: Taking out the greatest common factor from all terms.
• **Difference of Squares**: Used for polynomials in the form of \[ a^2 - b^2 = (a + b)(a - b) \]
• **Trinomial Factoring**: Factoring expressions of the form ax² + bx + c.
• **Grouping Method**: This technique is useful when factoring polynomials with four terms.

For the quadratic equation in the exercise, we used a combination of the coefficients and trinomial factoring.

Polynomial Coefficients

Understanding polynomial coefficients is crucial in factoring. Coefficients are the numerical part of the terms in a polynomial. For example, in 2x² + 3x + 1, the coefficients are:

• a = 2: coefficient of x².
• b = 3: coefficient of x.
• c = 1: constant term.

We manipulate these coefficients to rewrite the polynomial in a factorable form. Specifically, in the exercise, we used the coefficients to identify pairs of numbers that help in rewriting and factoring the polynomial using the coefficients' product (ac) and sum (b).

Grouping Method

The Grouping Method is a useful technique for factoring polynomials, especially when dealing with polynomials that have four terms. Here are the steps:

• **First**, we split the middle term into two terms such that their coefficients multiply to 'ac' and add up to 'b'.
• **Next**, we group the polynomial into two pairs. For example, we separated 2x² + 3x + 1 into (2x² + x) and (2x + 1).
• **Finally**, we factor out the common factor in each pair and then factor out the common binomial factor. In the exercise, the common binomial factor was (2x + 1), resulting in the factored form (2x + 1)(x + 1).

This method ensures the polynomial is simplified step by step, making it more manageable.