Problem 43 - Algebra 1: Concepts and Skills

Question

COMMON FACTOR Factor the expression. $$ 27 t^{2}+18 t+9 $$

Step-by-Step Solution

Answer

The factored form of the expression is \(9(3t^{2} + 2t + 1)\).

Step 1: Identify common factor in terms

The given expression is \(27t^{2} + 18t + 9\). Looking at the numbers associated with each term, we notice that 9 is a common factor.

Step 2: Factor out the common factor

Once we identified 9 as the common factor, we factorize it out of the expression. This gives us \(9(3t^{2} + 2t + 1)\).

Step 3: Simplify the quadratic expressions

The quadratic expression inside the parenthesis \(3t^{2} + 2t + 1\) cannot be factored further, as it does not have any more common factors. So, we accept this as the final factored form.

Key Concepts

Common Factor

Understanding the concept of a common factor is a fundamental step in simplifying polynomial expressions. A common factor is a number or expression that divides evenly into each term of a polynomial. To factor out a common factor, you need to identify the largest number or expression that can evenly divide each term in the polynomial.

For example, in the expression given as an exercise, \(27t^{2} + 18t + 9\), we look at the coefficients of each term: 27, 18, and 9. We determine the greatest common factor by finding the largest number that divides all these coefficients evenly, which is 9. By factoring out the 9, the expression simplifies significantly.

• Look at each term's coefficient.
• Find the largest number that divides all coefficients.
• Factor this number out of the expression.

Understanding this process helps in simplifying polynomial expressions efficiently.

Polynomial Expressions

Polynomial expressions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. They are foundational in algebra and appear in various forms, including linear, quadratic, cubic, and higher degree polynomials.

• Linear: Expressions like \(ax + b\).
• Quadratic: Expressions like \(ax^2 + bx + c\).
• Cubic or higher: Involving terms like \(ax^3 + bx^2 + cx + d\).

In our original exercise, the polynomial is a quadratic expression because the highest power of the variable \(t\) is 2. Polynomial expressions can often be factored to reveal their roots or simplify calculations. Understanding how to manipulate these expressions is key in algebraic problem-solving.

Quadratic Expressions

Quadratic expressions are a specific type of polynomial expression characterized by a degree of two. The standard form is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. Factoring quadratic expressions can sometimes reveal solutions to equations set to zero, also known as finding the roots or solutions.

In the exercise's quadratic expression \(3t^2 + 2t + 1\), we examined whether it could be factored further after factoring out the common factor. Since this expression does not have any evident further factorization (like two numbers that multiply to give the product of \(a\cdot c\) and add to \(b\)), we determine it is already in its simplest form. Knowing when a quadratic cannot be factored further is just as vital as mastering factoring itself.

• Recognize standard quadratic form \(ax^2 + bx + c\).
• Check if it can be factored further.
• Accept its simplest form when no factors are found.

This understanding helps in solving quadratic equations and expressing them in simplest terms.