Question

Find the least common multiple of the expressions. \(x^2-2 x-63, x+7\)

Short answer

The least common multiple (LCM) of the expressions \(x^2-2x-63\) and \(x+7\) is \((x+7)(x-9)\).

Step-by-step solution

Factorize the quadratic expression

The quadratic expression, \(x^2-2x-63\), can be factored into the form \(x-p)(x-q)\), where p and q are numbers. We need to look for two numbers that multiply to -63 (i.e., the constant term) and add up to -2 (i.e., the coefficient of the x term). After checking, we find -9 and 7 to be those numbers. Therefore, the quadratic expression can be factored into: \(x^2-2x-63 = (x-9)(x+7)\).

Determine the common factor

(x-9) and (x+7) are the factors of the original quadratic expression, \(x^2-2x-63\). By comparing these with the second expression, \(x+7\), we find that (x+7) is a common factor.

Determine the Least Common Multiple (LCM)

The least common multiple (LCM) of expressions is obtained by taking the product of the highest powers of all factors occurring in the given expressions. Here, (x+7) is the common factor and (x-9) is the additional factor from the first expression. Hence, the LCM would be the product of these two factors: \(LCM = (x+7)(x-9)\).

Key concepts

Factoring Quadratics

When dealing with quadratic expressions, one of the key techniques to master is factoring. A quadratic expression typically takes the form \(ax^2 + bx + c\). The goal is to break it down into simpler expressions that, when multiplied together, return the original quadratic. This is particularly useful when you need to solve equations or find common factors in algebraic expressions.

To factor a quadratic like \(x^2 - 2x - 63\), you should follow these steps:

• Identify the coefficients: \(a = 1\), \(b = -2\), and \(c = -63\).
• Find two numbers that multiply to \(a \times c = -63\) and add to \(b = -2\). These numbers are \(-9\) and \(7\).
• Rewrite the quadratic as \((x - 9)(x + 7)\).

Factoring quadratics is a foundational skill in algebra, helping simplify problems and find solutions more quickly. Practice makes perfect, so always try to factor on your own before relying on solutions.

Polynomial Expressions

Polynomial expressions, such as quadratics, are algebraic expressions that include variables raised to whole number powers. These can be as simple as a single term \(3x\) or as complex as multi-term expressions like \(x^3 + 2x^2 - x + 5\). Understanding how to work with polynomials is crucial in algebra.

In the context of our problem, the polynomial expression \(x^2 - 2x - 63\) was factored into \((x - 9)(x + 7)\). Polynomials can often be simplified through factoring, which may reveal shared factors with other expressions, as seen with \(x + 7\) in both factors. Simplifying polynomials can make operations like addition, subtraction, and finding least common multiples (LCM) much easier.

To get comfortable with polynomial expressions, practice expanding, simplifying, and factoring them whenever you can. This will build your confidence in handling more complex algebraic problems.

Algebraic Techniques

Algebraic techniques are the strategies and methods used to manipulate algebraic expressions to achieve a desired goal, like solving equations or simplifying expressions. In this exercise, we employed several algebraic techniques to find the Least Common Multiple (LCM) of two polynomial expressions.

The process involved:

• Factoring the quadratic expression \((x^2 - 2x - 63)\) into linear factors \((x-9)\) and \((x+7)\).
• Identifying common factors between expressions. In our case, \(x+7\) was a shared factor between \(x^2 - 2x - 63\) and \(x+7\).
• Combining these factors to determine the LCM as \((x+7)(x-9)\).

These algebraic techniques are invaluable in simplifying problems, allowing you to solve equations more easily and understand the relationships between different algebraic expressions. Regular practice with these techniques will sharpen your algebra skills, making it easier to tackle a wide range of mathematical challenges.