Problem 61 - Algebra 2

Question

What is the remainder when \(x^{2}-5 x+7\) is divided by \(x+1 ?\) \(\begin{array}{llll}{\text { A. }-13} & {\text { B. }-1} & {\text { C. } 1} & {\text { D. } 13}\end{array}\)

Step-by-Step Solution

Answer

The remainder when dividing the polynomial by x+1 is 13.

Step 1: Understand the Remainder Theorem

The Remainder Theorem states that for a polynomial function f(x), the remainder of the division of f(x) by (x - c) is f(c). Here, since we are dividing by (x + 1), c would be -1.

Step 2: Apply the Remainder Theorem

Substitute 'c' with -1 in the polynomial: f(x) = x^2 - 5x + 7 to get f(-1).

Step 3: Calculate the Value of f(-1)

Replace x with -1 in the polynomial to find the remainder: f(-1) = (-1)^2 - 5(-1) + 7.

Step 4: Simplify the Expression

Simplify the expression f(-1) = 1 + 5 + 7 to find the remainder.

Step 5: Find the Remainder

Add the values to find the remainder: 1 + 5 + 7 = 13.

Key Concepts

Polynomial Division

Polynomial division is a vital concept in algebra that involves dividing a polynomial by another polynomial. It is analogous to long division with numbers. Imagine you're dividing a pizza into equal parts; polynomial division does the same with algebraic expressions. The process is used to simplify expressions, solve polynomial equations, and find the remainder when dividing one polynomial by another, which ties directly into the Remainder Theorem. One uses polynomial division in cases where synthetic division is not applicable, such as when dividing by a polynomial of degree higher than one.

• Dividend: The polynomial you're starting with, akin to the whole pizza before cutting.
• Divisor: The polynomial you're dividing by, which sets the size of each slice.
• Quotient: The result of the division, representing the number of slices.
• Remainder: What's left over, just in case the pizza doesn't divide evenly.

When diving into polynomial division, remember it's an extension of what you learned with numbers, just with variables included.

Algebraic Long Division

Algebraic long division is a method used when dividing polynomials, similar to the long division process we use for numbers. It's the technique you would employ when synthetic division isn't applicable or when you need to see the details of the division process. Imagine dividing a multi-digit number by another number. In algebraic long division, you follow a similar set of steps, but with polynomials:

• Divide the first term of the dividend by the first term of the divisor to find the first term of the quotient.
• Multiply the entire divisor by this new term and subtract the result from the dividend.
• Bring down the next term of the dividend and repeat the process until you've worked through the entire dividend.

This method allows you to visualize each part of the process and understand how the parts of the polynomial are being divided, which can be helpful for complex divisions.

Synthetic Division

Synthetic division is a shorthand, faster method of polynomial division when you are dividing by a linear factor, i.e., a first-degree polynomial like (x - c). Its beauty lies in its simplicity and efficiency, especially beneficial for polynomials of higher degrees. Here's a brief overview of the steps:

• Write down the coefficients of the polynomial.
• Use the zero of the divisor (the value that makes the divisor equal to zero) opposite in sign.
• Bring down the leading coefficient to the bottom row.
• Multiply this leading coefficient by the zero of the divisor and write the result under the second coefficient, then add straight down.
• Continue this process of multiplying and adding down the line until you reach the end.

At the end, you'll have the coefficients of the quotient and the last number will be the remainder. Synthetic division is significantly less labor-intensive than algebraic long division, making it a favorite tool among students and educators alike.

Polynomial Functions

Polynomial functions are like the building blocks of algebra. They involve terms of variables raised to whole number exponents and their corresponding coefficients, all added together. The simplest example is a linear function, like y = mx + b, and they can get as complex as needed with higher degrees (where degree is the highest power of the variable in the polynomial). Characteristics of Polynomial Functions include:

• They are continuous, meaning there are no breaks, holes, or gaps in the graph.
• They have end behaviours depending on the leading term's coefficient and exponent.
• The degree of the polynomial gives the maximum number of turns the function's graph can have.
• The graph of a polynomial function will intersect the y-axis at the y-intercept, which is the constant term of the polynomial.

Whether you're solving an equation, finding a function's graph, or simply evaluating a point like in the Remainder Theorem, understanding polynomial functions is indispensable in algebra and calculus.