Problem 15 - Algebra for College Students

Question

Use synthetic division to determine the quotient and remainder for each problem. $$ \left(x^{3}+6 x^{2}-8 x+1\right) \div(x+7) $$

Step-by-Step Solution

Answer

Quotient: \( x^2 - x - 1 \), Remainder: 8.

Step 1: Set up synthetic division

Start by writing down the coefficients of the polynomial \( x^3 + 6x^2 - 8x + 1 \) which are \( 1, 6, -8, \) and \( 1 \). The zero of the divisor \( x+7 \) is \( -7 \). Use this zero for synthetic division.

Step 2: Begin the synthetic division process

Write the zero of the divisor, \( -7 \), to the left of a vertical bar. Write the coefficients \( 1, 6, -8, 1 \) to the right of the bar. Leave space below for writing results.

Step 3: Carry down the first coefficient

Carry down the first coefficient, \( 1 \), under the line as it is. This will be the leading coefficient of the quotient.

Step 4: Multiply and add

Multiply \( -7 \) by the number just carried down, \( 1 \), which gives \( -7 \). Write this below the next coefficient (\( 6 \)) and add, giving \( -1 \).

Step 5: Repeat the process

Multiply \( -7 \) by \( -1 \) giving \( 7 \). Write \( 7 \) below the next coefficient \( -8 \) and add, resulting in \( -1 \).

Step 6: Final multiplication and addition

Multiply \( -7 \) by \( -1 \) to get \( 7 \). Add this \( 7 \) to \( 1 \) (the last coefficient), which results in \( 8 \). This is the remainder.

Step 7: Interpret the results

The numbers below the line \( 1, -1, -1 \) represent the coefficients of the quotient polynomial, which is \( x^2 - x - 1 \), and the final number \( 8 \) is the remainder.

Key Concepts

Polynomial Division

Polynomial division is a fundamental concept in algebra that involves dividing one polynomial by another. It is similar to numerical long division, but instead, we deal with variables and coefficients. The goal is to determine a quotient polynomial and possibly a remainder polynomial.

When performing polynomial division, there are mainly two methods: long division and synthetic division. Long division is more general, allowing division by any polynomial with multiple terms. Synthetic division is a shortcut that simplifies the process when dividing by a linear polynomial, such as \(x+c\).

Steps in Polynomial Division:

• Identify the dividend and the divisor polynomials.
• Set up the division, arranging the dividend in standard form (descending powers of \(x\)).
• Apply the chosen method, either long division or synthetic division.
• Calculate the quotient and remainder if any residual is left.

In this exercise, we use synthetic division because the divisor is \(x+7\), a linear polynomial. Synthetic division simplifies the calculation process significantly.

Remainder Theorem

The Remainder Theorem is a helpful tool when dividing polynomials, particularly for finding remainders quickly. It states that the remainder of the division of a polynomial \(f(x)\) by a linear divisor \(x-a\) is equal to \(f(a)\).

In simpler terms, when you substitute \(a\) into the polynomial \(f(x)\), the result will be the remainder of the division. This theorem can save you time because you don't need to complete the entire division process to find the remainder.

In the current exercise, after performing synthetic division of \(x^3 + 6x^2 - 8x + 1\) by \(x + 7\), the remainder is \(8\). Using the theorem: You can find this by evaluating the polynomial at \(-7\), confirming the synthetic method:

Rational Root Theorem

The Rational Root Theorem provides a systematic way to find possible rational roots of a polynomial equation. It states that any rational solution, expressed as \(\frac{p}{q}\), is such that \(p\) is a factor of the constant term, and \(q\) is a factor of the leading coefficient.

For example, in the polynomial \(x^3 + 6x^2 - 8x + 1\), the constant term is \(1\) and the leading coefficient is \(1\). The only factor of \(1\) is \( \pm 1\). Therefore, the possible rational roots are \(+1\) and \(-1\). However, verifying potential roots must be done by further testing their validity within the polynomial.

This theorem is useful for predicting rational solutions and provides initial candidates to test when solving polynomial equations. In practice, the Rational Root Theorem complements synthetic division as it narrows down the potential roots to check.