Problem 151 - Elementary Algebra

Question

Factor. \(2 t^{2}+7 t+5\)

Step-by-Step Solution

Answer

\((2t + 5)(t + 1)\)

Step 1: Identify the quadratic form

Recognize that the given expression is a quadratic trinomial of the form \(ax^2 + bx + c\). Here, \(a = 2\), \(b = 7\), and \(c = 5\).

Step 2: Find two numbers that multiply to ac and add to b

Calculate the product of \(a\) and \(c\), which is \(2 \times 5 = 10\). We need to find two numbers that multiply to 10 and add to 7. These numbers are 2 and 5.

Step 3: Split the middle term using the identified numbers

Rewrite the middle term \(7t\) using the numbers found: \(2t + 5t\). The expression becomes \(2t^2 + 2t + 5t + 5\).

Step 4: Factor by grouping

Group the terms into two pairs: \((2t^2 + 2t) + (5t + 5)\). Factor out the GCF from each pair. From \(2t^2 + 2t\), factor out \(2t\), and from \(5t + 5\), factor out 5. The expression becomes \(2t(t + 1) + 5(t + 1)\).

Step 5: Factor out the common binomial

Factor out the common binomial factor \((t + 1)\), resulting in \((2t + 5)(t + 1)\).

Key Concepts

quadratic trinomials

A quadratic trinomial is a polynomial with three terms. The general form is represented as \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(x\) is the variable. The term \(ax^2\) is the quadratic term, \(bx\) is the linear term, and \(c\) is the constant term.

Understanding these terms is crucial because it helps in identifying the values needed for factoring. For example, in \(2t^2 + 7t + 5\), the quadratic term is \(2t^2\), the linear term is \(7t\), and the constant term is \(5\).

Identifying these components allows us to apply techniques like factoring by grouping to simplify the polynomial.

factoring by grouping

Factoring by grouping is an effective method for simplifying polynomials. The process involves breaking down a polynomial into groups that can be easily factored.

Steps to Factor by Grouping:

• Split the polynomial into two groups: For example, split \(2t^2 + 7t + 5\) as \(2t^2 + 2t\) and \(5t + 5\).
• Factor out the Greatest Common Factor (GCF) from each group: For \(2t^2 + 2t\), the GCF is \(2t\). For \(5t + 5\), the GCF is \(5\).

After factoring out the GCF from each group, you get \(2t(t + 1)\) and \(5(t + 1)\).

Next, factor out the common binomial, \(t + 1\). The final factored form is \( (2t + 5)(t + 1) \).

polynomial factorization

Polynomial factorization involves expressing a polynomial as a product of its factors. This helps in simplifying and solving polynomial equations.

For our example, the polynomial \(2t^2 + 7t + 5\) can be factored by identifying that it’s a quadratic trinomial and then using techniques such as factoring by grouping.

Here are the steps in brief:

• Identify the quadratic trinomial: \(ax^2 + bx + c\).
• Find two numbers that multiply to \(ac\) and add to \(b\).
• Rewrite the middle term using these numbers and group the terms.
• Factor out the GCF of each group.
• Lastly, factor out the common binomial.

The factored form of \(2t^2 + 7t + 5\) is \( (2t + 5)(t + 1) \). This method makes it easier to solve equations and understand polynomial behaviors.