Problem 50 - Algebra A Combined Function

Question

Factor each trinomial by grouping. Exercises 9 through 12 are broken into parts to help you get started. $$ 5-12 x+7 x^{2} $$

Step-by-Step Solution

Answer

The factored form is \((7x - 5)(x - 1)\).

Step 1: Rewrite the Trinomial

Rewrite the trinomial in standard form, where the terms are in descending order of exponent.\[ 7x^2 - 12x + 5 \]

Step 2: Identify the Product and Sum

Identify two numbers that multiply to give the product of the quadratic coefficient and the constant term (\(7 \times 5 = 35\)) and add to give the linear coefficient (-12).Find two numbers: -5 and -7.

Step 3: Break Down the Middle Term

Use the two numbers identified in step 2 to split the middle term (-12x) into two separate terms.\[ 7x^2 - 7x - 5x + 5 \]

Step 4: Group the Terms

Group the terms into pairs:\[ (7x^2 - 7x) + (-5x + 5) \]

Step 5: Factor Each Group

Factor out the greatest common factor from each group:\[ 7x(x - 1) - 5(x - 1) \]

Step 6: Factor by Grouping

Notice that \(x - 1\) is common in both groups. Factor this common binomial factor out.\[ (7x - 5)(x - 1) \]

Step 7: Verify the Factored Form

Verify your solution by expanding the factors to ensure they equal the original trinomial.\[ (7x - 5)(x - 1) = 7x^2 - 7x - 5x + 5 = 7x^2 - 12x + 5 \]

Key Concepts

Quadratic Equations

Quadratic equations are a central concept in algebra and are represented in the standard form as \[ ax^2 + bx + c = 0 \]. Here, "a," "b," and "c" are constants, with "a" not equal to zero. The term with \( x^2 \) is what makes the equation quadratic.
These types of equations can describe parabolic curves on a graph. They are widely used in various fields such as physics, engineering, and economics.
When factoring quadratic equations, especially in the context of trinomials like the exercise mentioned, it involves rewriting the quadratic expression as a product of two simpler binomial expressions. This method is also known as solving by factoring.
Factoring helps simplify the equations and find the roots or solutions, i.e., the values of "x" that satisfy the equation.

• Roots make each binomial equal to zero when solved separately.
• The quadratic can be completely factored if two numbers can be found that multiply to the product of "a" and "c" and add to "b."

Polynomial Expressions

Polynomials are expressions made up of variables and constants combined using addition, subtraction, multiplication, and non-negative integer exponents.
A trinomial is a special kind of polynomial that has exactly three terms. In the exercise above, the polynomial expression, \( 7x^2 - 12x + 5 \), is a trinomial.
Understanding polynomials, especially trinomials, is essential in various algebraic techniques, including factoring.

• They are often used in equations to represent relationships and properties.
• When working with trinomials, the strategy often focuses on breaking down or rearranging these three terms to simplify or solve equations efficiently.

The key to working with any polynomial expression is to ensure it is in standard form. This means arranging the terms by descending degree.
For trinomials, this generally makes it easier to identify relationships between the terms, such as the products or the sums of coefficients needed for factoring.

Algebraic Techniques

Algebraic techniques are powerful tools used to manipulate, solve, and simplify expressions and equations. One of the most common techniques employed in solving quadratic equations is factoring by grouping, as illustrated in the original exercise.
Factoring by grouping involves several systematic steps.

• First, identify a way to separate the middle term into two terms whose coefficients multiply to the same product as the leading and constant term when combined.
• Next, rearrange and group terms into pairs where a common factor can be identified.
• Factoring out the greatest common factor from each group simplifies the expression further.

In the final step, it involves recognizing a common binomial factor, as seen with the expression \( 7x(x - 1) - 5(x - 1) \), which can then be factored to \( (7x - 5)(x - 1) \).
This technique not only helps in solving complex expressions but also enhances your overall algebraic problem-solving skills. By practicing these methods, you can approach, break down, and factor expressions effectively, making algebra less daunting.