Problem 22 - Algebra 2 and Trigonometry

Question

In \(9-26,\) write each expression as the product of two binomials. $$ 2 y^{2}+5 y-3 $$

Step-by-Step Solution

Answer

The product of two binomials is \((2y - 1)(y + 3)\).

Step 1: Identify Terms and Structure

The given expression is a quadratic trinomial of the form \( ax^2 + bx + c \), where \( a = 2 \), \( b = 5 \), and \( c = -3 \). Our goal is to turn this expression into a product of two binomials.

Step 2: Multiply and Find Two Numbers

Multiply the leading coefficient \( a = 2 \) by the constant term \( c = -3 \) to get \( 2 \times -3 = -6 \). Find two numbers that multiply to \(-6\) and add to \(5\), the middle coefficient. These numbers are \(6\) and \(-1\), as \( 6 \times -1 = -6 \) and \( 6 + (-1) = 5 \).

Step 3: Rewrite Middle Term

Use the pair of numbers from Step 2 to rewrite the middle term. The expression \(2y^2 + 5y - 3\) becomes \(2y^2 + 6y - y - 3\).

Step 4: Group and Factor

Group the terms into pairs: \((2y^2 + 6y) + (-y - 3)\). Factor out the greatest common factor in each group. From the first group, factor out \(2y\) to get \(2y(y + 3)\). From the second group, factor out \(-1\) to get \(-1(y + 3)\).

Step 5: Create Binomials

Notice that both groups contain the common binomial \((y + 3)\). Factor this common binomial out to get \((2y - 1)(y + 3)\). This product of binomials is the factored form of the original expression.

Key Concepts

Binomials

In algebra, a binomial is simply an expression with two terms. Each of these terms can contain numbers, variables, or both. Think of binomials as the building blocks of many algebraic expressions, particularly when working with polynomials. For instance, an example of a binomial is \( y + 3 \). The two components, 'y' and '3', are added together.Understanding binomials is crucial when factoring larger expressions, like quadratic trinomials. The goal is to express a quadratic trinomial as a product of two simpler binomials. This process can simplify solving equations and understanding relationships between variables. By focusing on binomials, you're essentially breaking down a complex expression into smaller and more manageable parts.

Quadratic Expressions

Quadratic expressions are polynomials of degree two, typically represented as \( ax^2 + bx + c \). The key characteristic is that the highest exponent of the variable is 2. This makes quadratics very special because they form parabolas when graphed.Let's take the expression \( 2y^2 + 5y - 3 \), the one we tackle in the exercise. Here, our task is to break down this quadratic into simpler components. Moreover, every quadratic equation can be factored into the product of two binomials. This means translating the expression into simpler terms, which can aid in solving or analyzing various mathematical problems.

Factoring Techniques

Factoring quadratic trinomials like \( 2y^2 + 5y - 3 \) requires a strategic approach. One effective method is to find two numbers that multiply to the product of the leading coefficient and the constant term, while also adding up to the middle coefficient.For example:

• Multiply the leading coefficient, '2', by the constant term, '-3'. This equals '-6'.
• Next, find two numbers that multiply to '-6' and add up to '5', the middle coefficient. The numbers '6' and '-1' fit perfectly, as \( 6 \times -1 = -6 \) and \( 6 + (-1) = 5 \).

After determining these numbers, use them to rewrite the quadratic expression by splitting the middle term. Once that's done, use grouping and further factorization to get to the binomial form. The ultimate goal is to make the expression easily solvable, which helps in understanding and manipulating it for various algebraic tasks.