Problem 208 - Elementary Algebra

Question

Factor. \(2 b^{2}-7 b+4\)

Step-by-Step Solution

Answer

\( (b - 4)(2b - 1) \)

Step 1: Setup for factoring

Identify a, b, and c in the quadratic expression in the form of \( ax^2 + bx + c \).Here, a = 2, b = -7, c = 4.

Step 2: Calculate the product of a and c

Multiply a and c: \( 2 \times 4 = 8 \). We need to find two numbers that multiply to 8 and add to -7.

Step 3: Identify the two numbers

The two numbers that satisfy the conditions are -1 and -8. \( -1 \times -8 = 8 \) and \( -1 + (-8) = -7 \).

Step 4: Rewrite the middle term

Rewrite the quadratic expression by splitting the middle term using the identified numbers: \( 2b^2 - b - 8b + 4 \).

Step 5: Factor by grouping

Group the terms to factor separately: \( (2b^2 - b) + (-8b + 4) \). Factor out the greatest common factor from each group: \( b (2b - 1) - 4 (2b - 1) \).

Step 6: Factor out the common binomial

Factor out the common binomial factor, which is \( 2b - 1 \): \( (b - 4)(2b - 1)\).

Key Concepts

quadratic equations

Quadratic equations are polynomial equations of degree two. This means the highest power of the variable in the equation is squared. They take the general form: \[ ax^2 + bx + c = 0 \]. In this format, \(a\), \(b\), and \(c\) are coefficients where \(a ≠ 0\). Common examples include expressions like \(2b^2 - 7b + 4\) that we see in the exercise. Quadratic equations can be solved by different methods, such as:

• Factoring
• Completing the square
• Using the quadratic formula

Factoring is often the first method to try, as it breaks down the quadratic into simpler, solvable parts. This method is effective when the equation can be rewritten as the product of two binomial factors.

factoring by grouping

Factoring by grouping is a helpful technique when dealing with polynomials. This method is effective when a polynomial can be arranged into groups that have common factors. In the given exercise, the expression \(2b^2 - 7b + 4\) is factored by grouping:

1. First, rewrite the middle term as the sum of two terms whose coefficients add up to the middle coefficient and multiply to the product of \(a\) and \(c\).
2. Next, group the terms: \((2b^2 - b) + (-8b + 4)\).
3. Finally, factor out the greatest common factor from each group: \(b(2b - 1) - 4(2b - 1)\).

Notice here that \(2b - 1\) is a common binomial factor. Factoring it out further simplifies the expression to \((b - 4)(2b - 1)\). This process turns a complicated polynomial into a product of simpler binomials.

polynomial factorization

Polynomial factorization involves breaking down a polynomial into the product of smaller polynomials or factors. For quadratic expressions, this often means rewriting the polynomial as a product of two binomials. The steps generally include:

• Identifying the coefficients \(a\), \(b\), and \(c\).
• Finding two numbers that multiply to \(a \times c\) and add to \(b\).
• Rewriting the middle term using these two numbers.
• Using factoring by grouping to factor out the common binomial factor.

For our given example \(2b^2 - 7b + 4\), we identified the numbers -1 and -8, which satisfy the conditions. Then, we rewrote the expression, grouped it, and factored it ultimately into \((b - 4)(2b - 1)\). Polynomial factorization simplifies the equation and allows for easier solutions.

algebraic expressions

Algebraic expressions consist of variables, coefficients, and operations such as addition, subtraction, multiplication, and division. Polynomial expressions are a subset of algebraic expressions that are made up of terms connected by these operations. Quadratic expressions, like \(2b^2 - 7b + 4\), specifically involve terms up to the second degree. Manipulating algebraic expressions involves various techniques such as:

• Expanding
• Simplifying
• Factoring

By breaking down and reorganizing these expressions, we can solve equations and understand relationships between variables more easily. Knowing how to work with algebraic expressions is fundamental in algebra and crucial for solving quadratic equations.