Problem 19 - Elementary and Intermediate Algebra

Question

Factor. See Example 1 or Example 6. $$ 2 x^{2}+3 x+1 $$

Step-by-Step Solution

Answer

The expression \(2x^2 + 3x + 1\) factors to \((2x + 1)(x + 1)\).

Step 1: Identify the Coefficients

The quadratic expression is in the form \( ax^2 + bx + c \). Here, \( a = 2 \), \( b = 3 \), and \( c = 1 \). This helps us in looking for factors that satisfy the factored form.

Step 2: Multiply the Leading Coefficient and Constant

Multiply the leading coefficient \( a = 2 \) by the constant term \( c = 1 \). This gives us \( 2 imes 1 = 2 \). Our task is to find two numbers that multiply to this product (2) and add up to the middle coefficient \( b = 3 \).

Step 3: Find the Numbers that Satisfy the Conditions

We need two numbers that multiply to 2 and add to 3. The numbers 2 and 1 satisfy these conditions since \( 2 imes 1 = 2 \) and \( 2 + 1 = 3 \).

Step 4: Rewrite the Middle Term

Rewrite the middle term \( 3x \) as \( 2x + 1x \) using the numbers found in Step 3. The expression becomes: \[ 2x^2 + 2x + 1x + 1 \].

Step 5: Factor by Grouping

Group the terms into two pairs: \( (2x^2 + 2x) + (1x + 1) \). Factor out the common factors in each group. This gives: \[ 2x(x + 1) + 1(x + 1) \].

Step 6: Factor the Common Binomial

Notice that \((x + 1)\) is a common factor. Factor it out: \[ (2x + 1)(x + 1) \].

Step 7: Verify the Factored Form

Expand \((2x + 1)(x + 1)\) to check if it equals the original expression. \( 2x imes x + 2x imes 1 + 1 imes x + 1 imes 1 = 2x^2 + 2x + 1x + 1 = 2x^2 + 3x + 1 \). It matches, so the factorization is correct.

Key Concepts

Quadratic Expressions

Quadratic expressions are algebraic expressions that include terms up to the second degree. The general form is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. The term with the highest power of \(x\) is \(ax^2\), which makes it a quadratic expression because it results in a parabola when graphed. This particular equation holds critical significance in algebra because it introduces the concept of polynomials being expressed and manipulated. Working with quadratic expressions is fundamental in understanding how these equations are used to describe parabolic paths in real-world scenarios, like the motion of projectiles.

Factoring by Grouping

Factoring by grouping is a clever technique used to factor polynomials. This method involves rearranging and grouping terms to reveal common factors, making it easier to simplify expressions. The idea is to split a polynomial into manageable groups with common terms. In the example \(2x^2 + 3x + 1\), it is rewritten as \(2x^2 + 2x + 1x + 1\) by using the middle term to help form two groups: \((2x^2 + 2x)\) and \((1x + 1)\).

• Identify pairs within the expression that can be grouped together.
• Factor common elements out of each group.
• Look for a common binomial that can be factored out from the entire expression.

This technique helps to simplify and break down more complex algebraic expressions into their factors, which can then be more easily interpreted or solved. Understanding factoring by grouping is essential for grasping more intricate algebraic solutions.

Algebraic Expressions

Algebraic expressions form the backbone of algebra, allowing us to represent numbers and operations in a symbolic way. These expressions consist of variables, coefficients, and constants, like \(2x^2 + 3x + 1\), and they can include operations such as addition, subtraction, multiplication, and division. They serve as mathematical phrases that can articulate a variety of real-world situations or theoretical questions.

• A coefficient is the numerical factor that multiplies a variable.
• A constant is a fixed value that does not contain any variable component.
• Variables act as placeholders for numbers that can change.

Such expressions are foundational in many areas of mathematics, enabling us to solve equations, perform operations, and express relationships among quantities. Mastering the manipulation of algebraic expressions, like factoring, is a key skill in progressing through mathematics.