Problem 56 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ 3 x^{2}-4 x+2 $$

Step-by-Step Solution

Answer

The polynomial factors as \((x - 1)(3x - 2)\).

Step 1: Identify the Coefficients

The polynomial given is in the form \( ax^2 + bx + c \). Here, \( a = 3 \), \( b = -4 \), and \( c = 2 \). Identifying the coefficients is crucial for factoring the quadratic expression.

Step 2: Use the AC Method

Multiply the leading coefficient \( a \, (3) \) and the constant term \( c \, (2) \) to get \( 6 \). We need to find two numbers that multiply to \( 6 \) and add to \( -4 \). These numbers are \( -2 \) and \( -3 \).

Step 3: Rewrite the Middle Term

Rewrite the middle term \( -4x \) as \( -2x - 3x \). This changes the expression to: \( 3x^2 - 2x - 3x + 2 \).

Step 4: Group Terms

Group the first two terms and the last two terms: \((3x^2 - 2x) + (-3x + 2)\).

Step 5: Factor by Grouping

Factor out the greatest common factor from each group. From \(3x^2 - 2x\), factor out \(x\) to get \(x(3x - 2)\). From \(-3x + 2\), factor out \(-1\) to get \(-1(3x - 2)\).

Step 6: Factor Out the Common Binomial Factor

Both groups contain the common factor \(3x - 2\). Factor this out to write the expression as \((x - 1)(3x - 2)\).

Step 7: Verify the Solution

Multiply \((x - 1)(3x - 2)\) to check if it equals the original polynomial: \((x)(3x) = 3x^2\), \(-(1)(3x) = -3x\), \((x)(-2) = -2x\), and \(-(1)(-2) = 2\). Combine the terms: \(3x^2 - 4x + 2\), which is the original polynomial.

Key Concepts

Quadratic Expressions

A quadratic expression is a kind of polynomial that specifically involves terms up to the second degree. In its simplest form, a quadratic expression can be represented as \( ax^2 + bx + c \). This includes:

• \( ax^2 \): the quadratic term, which is the highest degree term.
• \( bx \): the linear term, which involves the first degree.
• \( c \): the constant term, which doesn't involve variables.

Quadratic expressions often appear in various real-world scenarios, such as calculating areas and solving problems involving projectile motion.
To factor these expressions, which means expressing them as a product of two binomials, one must understand the arrangement of these terms. Identifying each component and their coefficients sets the groundwork for the next steps in factoring.

Factor by Grouping

Factoring by grouping is a handy technique used to simplify polynomials, especially when dealing with quadratic expressions that don’t have a leading coefficient of 1. The core idea is to rearrange the terms and group them in pairs to make factoring easier.
Here's how it typically works with our quadratic expression:

• First, you divide the expression into two smaller groups.
• Next, factor out the greatest common factor (GCF) from each group.

For example, take the expression \( 3x^2 - 2x - 3x + 2 \):

• Split into \( (3x^2 - 2x) + (-3x + 2) \).
• From \( 3x^2 - 2x \), factor out an \( x \): yields \( x(3x - 2) \).
• From \( -3x + 2 \), factor out \( -1 \): yields \( -1(3x - 2) \).

Finally, you’ll notice a common binomial \( (3x - 2) \) that you can factor out, leading you to the simplified version of the expression. This method, while initially seeming more complex, can make challenging polynomials much more approachable.

AC Method

The AC Method is a strategic approach for factoring quadratic polynomials, particularly when the leading coefficient isn’t 1. The name stems from multiplying the 'A' part (the coefficient of the squared term) and the 'C' part (the constant term).
Here's a step-by-step way to use the AC Method:

• Multiply the leading coefficient \( a \) by the constant term \( c \). In our exercise, \( 3 \times 2 = 6 \).
• Identify two numbers that both add up to the middle coefficient \( b \) and multiply to \( ac \). Here, these numbers are \( -2 \) and \( -3 \), since \( -2 + -3 = -5 \) and \( -2 \, \times \, -3 = 6 \).

These steps prepare you for the grouping process, allowing you to rewrite the original quadratic expression in a factorable form. This method breaks down complex expressions, making them simpler to manage and solve.

Polynomial Coefficients

Coefficients are the numerical factors in front of the variables in a polynomial. Recognizing these is a key step in factoring and solving polynomials. In the expression \( ax^2 + bx + c \), each variable has a coefficient of:

• \( a \): the coefficient of \( x^2 \).
• \( b \): the coefficient of \( x \).
• \( c \): the constant coefficient.

In our example, we have \( a = 3 \), \( b = -4 \), and \( c = 2 \).
These coefficients give the polynomial its shape and behavior. They indicate the degree's intensity and direction. Recognizing coefficients early is essential since they form the backbone of many factoring techniques, including the AC Method and grouping.
By closely inspecting these numbers, students can unlock efficient pathways to decompose and solve even the most intimidating polynomial expressions.