Question

Factor each polynomial. $$ 3 x^{2}-14 x+8 $$

Short answer

(3x - 2)(x - 4)

Step-by-step solution

Identify coefficients

Given the polynomial equation 3x^2 - 14x + 8 Identify the coefficients of the quadratic equation ax^2 + bx + c. Here, the coefficient a is 3, b is -14, and c is 8.

Find two numbers

Find two numbers that multiply to the product of a and c (3 * 8 = 24) and add to b (-14). We look for two numbers that multiply to 24 and add to -14. These numbers are -12 and -2.

Rewrite the middle term

Rewrite the polynomial by breaking up the middle term -14x into two terms using the numbers found in Step 2. 3x^2 - 12x - 2x + 8

Factor by grouping

Group the terms in pairs: (3x^2 - 12x) - (2x - 8) Factor out the greatest common factor (GCF) from each pair: 3x(x - 4) - 2(x - 4)

Factor out the common binomial factor

Notice that (x - 4) is a common factor. Factor (x - 4) out of each term: (3x - 2)(x - 4)

Key concepts

quadratic equations

Quadratic equations are polynomials of degree 2, meaning the highest exponent of the variable (usually represented as x) is 2. They are commonly written in the form: \[ax^2 + bx + c = 0\]
Here, \(a\), \(b\), and \(c\) are coefficients, and \(a\) should not equal zero because if \(a\) is zero, then the equation becomes linear, not quadratic. The graph of a quadratic equation is a parabola that can open upwards or downwards depending on the sign of the coefficient \(a\).
To solve quadratic equations, there are several methods such as factoring, using the quadratic formula, completing the square, or graphing.

coefficients

Coefficients are numerical values that multiply the variables in an algebraic expression. In the context of quadratic equations, the standard form \(ax^2 + bx + c\) includes three coefficients:

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

In our example, \(3x^2 - 14x + 8\):

• \(a = 3\)
• \(b = -14\)
• \(c = 8\)

These coefficients help us in various methods of solving quadratic equations. Identifying them correctly is the first step in the factoring process.

factor by grouping

Factoring by grouping is a technique used to factor polynomials that have four or more terms. This method involves a few steps:
First, split the polynomial into groups that have common factors. Next, factor out the greatest common factor (GCF) from each group. Finally, if each group contains a common binomial factor, you can factor this binomial out. Here’s how it applies to our example: \(3x^2 - 12x - 2x + 8\) gets grouped as follows:

• Group terms: \((3x^2 - 12x) - (2x - 8)\)
• Factor each group: \(3x(x - 4) - 2(x - 4)\)
• Factor the common binomial: \((3x - 2)(x - 4)\)

This process simplifies the polynomial into a product of two binomials.

common binomial factor

A common binomial factor is a binomial that can be factored out from multiple terms. In the Factor by Grouping method, you often end up with a binomial that appears across the terms you grouped.
In our example, after grouping and factoring, we obtained: \(3x(x - 4) - 2(x - 4)\). Notice how \((x - 4)\) appears in both terms. Since \((x - 4)\) is common, we can factor it out: \[(3x - 2)(x - 4)\]
Finding and factoring out a common binomial factor simplifies equations, making it easier to solve or further manipulate the expression. This is a crucial step in the factoring process and greatly aids in recognizing patterns in quadratic equations.