Question

Divide and simplify. $$a^{2}-15 a+56 \text { by } a-7$$

Short answer

The simplified form of the expression is (a-8).

Step-by-step solution

Set Up the Division

Write down the division as a fraction, with the polynomial numerator ((a^{2}-15a+56)) and the linear denominator ((a-7)). The expression should look like \[\frac{a^{2}-15a+56}{a-7}.\]

Factor the Numerator

Factor the quadratic polynomial in the numerator. Look for two numbers that multiply to give the constant term (+56) and add to give the coefficient of the middle term (-15). These numbers are -7 and -8, so factor the numerator to get \[(a-7)(a-8).\]

Cancel Common Factors

Cancel the common factor of ((a-7)) from the numerator and the denominator. The resulting expression will be \[\frac{(a-7)(a-8)}{a-7} = a-8.\]After cancellation, the simplified form of the polynomial division is (a-8).

Key concepts

Simplifying Algebraic Expressions

Understanding how to simplify algebraic expressions is an essential skill for tackling a wide array of mathematical problems. Simplification often involves condensing an expression into its most concise form, making it easier to work with.
Consider the original polynomial division problem: \[\frac{a^{2}-15a+56}{a-7}.\]
In the context of simplification, you begin by expressing the numerator as a fraction, identifying the highest common factors between the numerator and the denominator, then reducing the expression to its simplest form. For instance, in the numerator of our example \(a^{2}-15a+56\), we look to rewrite it in such a way that any common factors with the denominator become apparent.

Factoring Quadratics

Factoring quadratics is a central concept in algebra, serving as a foundation for simplifying algebraic expressions and solving equations. Quadratic equations are polynomials of the second degree, generally in the form \(ax^2 + bx + c \).
When facing a quadratic like \( a^{2}-15a+56 \), the goal is to break it down into the product of two binomials. This step is crucial before you divide by a linear factor. To factor, you look for two numbers that, when multiplied, give the constant term (in this case, +56) and, when added, provide the linear coefficient (-15). Here, these numbers are -7 and -8, leading to the factored form \((a-7)(a-8)\).
Factoring not only prepares us for simplifying expressions by division but also is a key step in other areas, such as solving quadratic equations and analyzing function graphs.

Cancelling Common Factors

Cancelling common factors is almost like finding a shortcut through the complicated forest of algebraic equations. It allows us to 'divide out' the same factor from the numerator and the denominator, simplifying the expression without altering its value.
In our exercise, after factoring, we identify \((a-7)\) as a common factor in both the numerator and denominator. Because anything divided by itself equals one, we can cancel this common factor. The process looks like this: \[\frac{(a-7)(a-8)}{a-7} = a-8.\]
Once we cancel \((a-7)\), we are left with \(a-8\), the simplest form of the original expression. This method is not just for polynomials; it’s used across various areas of mathematics, from reducing fractions to solving complex rational expressions. It's a powerful tool, simplifying problems and helping to prevent mistakes during calculations.