Problem 32 - College Algebra

Question

For the following exercises, use synthetic division to find the quotient. $$ \left(x^{4}-10 x^{3}+37 x^{2}-60 x+36\right) \div(x-2) $$

Step-by-Step Solution

Answer

The quotient is \(x^3 - 8x^2 + 21x - 18\).

Step 1: Setup Synthetic Division

To perform synthetic division, first write down the coefficients of the polynomial \(x^4 - 10x^3 + 37x^2 - 60x + 36\). These coefficients are \([1, -10, 37, -60, 36]\). Also, write the zero of the divisor \(x - 2\), which is \(x = 2\).

Step 2: Begin the Synthetic Division Process

Draw a horizontal line below the coefficients with space for another row. Write \(2\) on the left side of this setup, which is the root of \(x-2\). Write the first coefficient \(1\) directly beneath itself on a separate row.

Step 3: Perform the Division Steps

1. Multiply \(2\) by the number written below the line, \(1\), and write the product \(2\) under the \(-10\) (the second coefficient).2. Add \(-10\) and \(2\) to get \(-8\), and write this result below the line.3. Repeat this process: multiply \(2\) by \(-8\) to get \(-16\), write it beneath \(37\), add \(37 + (-16) = 21\), and write this below the line.4. Multiply \(2\) by \(21\) to get \(42\), write it beneath \(-60\), add: \(-60 + 42 = -18\), and write \(-18\) below.5. Multiply \(2\) by \(-18\) to get \(-36\), write this beneath \(36\), and add: \(36 + (-36) = 0\).

Step 4: Interpret the Result

The bottom row, after performing synthetic division, reads \([1, -8, 21, -18, 0]\). The last number, \(0\), is the remainder. The rest form the coefficients of the quotient: \(x^3 - 8x^2 + 21x - 18\).

Step 5: Write the Quotient

As a result, the quotient of the division is \(x^3 - 8x^2 + 21x - 18\). Since the remainder is zero, the divisor \(x-2\) is a factor of the polynomial.

Key Concepts

Polynomial Division

Polynomial division is a crucial process in algebra that resembles long division, but it's used for dividing polynomials by other polynomials. This method helps to simplify expressions, find zeros of polynomials, and solve polynomial equations.
Here's how it works:

• Set up the division by aligning the polynomials just like long division with numbers.
• Use the leading term of the divisor to divide the leading term of the dividend. This gives the first term of the quotient.
• Multiply the divisor by this first term of the quotient, subtract the result from the dividend, and repeat the process with the remainder.

An efficient variant of polynomial division is synthetic division, especially useful when dividing by a linear polynomial. It's more straightforward and involves fewer calculations, making it easier to find the quotient and remainder quickly.

Factor Theorem

The Factor Theorem is a key concept in algebra that connects factors and roots of polynomials. According to this theorem, a polynomial has a factor \((x - c)\) if and only if the polynomial evaluates to zero when \(x = c\).
This means:

• If a polynomial \(f(x)\) is divided by \((x - c)\) and the remainder is zero, then \((x - c)\) is a factor of \(f(x)\).
• If you find that substituting \(x = c\) into the polynomial yields zero, \(c\) is a root of the polynomial.

In our exercise, since the remainder was zero, \((x - 2)\) is a factor of the polynomial \(x^4 - 10x^3 + 37x^2 - 60x + 36\). This makes the Factor Theorem incredibly useful for simplifying polynomials and identifying their roots.

Remainder Theorem

The Remainder Theorem is a helpful mathematical tool for determining the remainder when dividing a polynomial by a linear divisor \((x - c)\). It states that the remainder of this division is simply the value of the polynomial evaluated at \(x = c\).
In simpler terms:

• Find \(f(c)\) where \(f(x)\) is the polynomial and \(c\) is the constant from the divisor \((x - c)\).
• The value \(f(c)\) is the remainder.

In our exercise, using synthetic division, we found that when \(x^4 - 10x^3 + 37x^2 - 60x + 36\) is divided by \(x - 2\), the remainder is zero. Using the Remainder Theorem, since \(f(2) = 0\), this confirms \(2\) is a root of the polynomial.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. Understanding how to manipulate these expressions is pivotal in solving equations and simplifying polynomials.
Basic components of algebraic expressions include:

• Terms: Separated by plus or minus signs, composed of constants, variables, and coefficients.
• Coefficients: Numbers multiplied by the variable(s) in the expression.
• Constants: Numbers on their own, without variables.

Expressions become polynomials when they consist entirely of terms that are combined using addition, subtraction, and multiplication. By learning how to handle polynomials through techniques like synthetic division, students can simplify complex expressions efficiently and solve higher-order equations.