Problem 104 - Algebra 1: Concepts and Skills

Question

Factor the trinomial. (Lessons 10.5,10.6) $$ 3 x^{2}-15 x+18 $$

Step-by-Step Solution

Answer

The trinomial \(3x^2 - 15x + 18\) factors to \(3(x-2)(x-3)\).

Step 1: Review the trinomial

The given trinomial is \(3x^2 - 15x + 18. \) It can be factored using the formula for factoring trinomials: \(ax^2 + bx + c = a(x-h)(x-k)\), where \(h\) and \(k\) are numbers that satisfy the equation \(hk = ac\) and \(h + k = -b/a\).

Step 2: Find 'h' and 'k'

We have \(a = 3\), \(b = -15\), and \(c = 18\). We need to find 'h' and 'k' such that \(hk = 3*18 = 54\) and \(h + k = -(-15)/3 = 5\). From factors of 54 (1,2,3,6,9,18,27, and 54), we see that 'h' and 'k' are 6 and 9.

Step 3: Factor the trinomial

Use the values of 'h' and 'k' to rewrite the trinomial as a product of two binomials. The trinomial \(3x^2 - 15x + 18\) then factors to \(3(x-2)(x-3)\).

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of the second degree that typically take the form \( ax^2 + bx + c = 0 \). These equations can include any real number coefficients, where \( a \), \( b \), and \( c \) are constants, and \( a \) cannot be zero. If \( a \) were zero, the equation would no longer be quadratic but linear.
The solutions to these equations are the "roots," which can be found using various methods such as factoring, the quadratic formula, or completing the square. Understanding quadratic equations is essential, as they appear frequently in mathematics and can model various real-world scenarios such as projectile motions and area problems.

Factoring Techniques

Factoring is a critical algebraic method used to simplify expressions, solve quadratic equations, and rewrite polynomials in a different form. The main goal of factoring is to express a polynomial as a product of simple polynomials or binomials. This process helps in identifying the solutions (or roots) of an equation, making it quicker and easier to solve.
When working with trinomials like \( ax^2 + bx + c \), the goal is to find two binomials that multiply to give the original trinomial. The formula \( ax^2 + bx + c = a(x-h)(x-k) \) helps in finding values for \( h \) and \( k \). For example, if we have \( 3x^2 - 15x + 18 \), finding integers \( h \) and \( k \) that satisfy \( hk = ac \) and \( h+k = -b \) is key. In this case, \( h \) and \( k \) are 6 and 9, resulting in the factored form \( 3(x-2)(x-3) \).

• Identify \( a \), \( b \), and \( c \) from the trinomial.
• Calculate \( ac \) and solve for \( hk = ac \) with \( h+k = -b/a \).
• Rewrite the original trinomial into a factored form based on \( h \) and \( k \).

Algebraic Expressions

Algebraic expressions consist of variables, numbers, and operations arranged in a mathematical sentence. They don’t have an equal sign like equations but are significant in forming equations and inequalities.
Factoring plays a crucial role in simplifying algebraic expressions, which allows for easier manipulation and understanding of expressions. For instance, given \( 3x^2 - 15x + 18 \), by factoring, we can break it down into simpler expressions, making further operations like solving or substituting easier. This is why understanding factoring techniques is essential, as it enables tackling more complex problems in diverse areas of mathematics and science.

• Recognize the structure of an expression and the operations within it.
• Use factoring to simplify and make expressions manageable.
• Apply simplified expressions in solving equations or real-world problems.

Understanding these elements will facilitate working with various algebraic challenges, ensuring a robust foundational knowledge for further math courses.