Problem 51 - Algebra for College Students

Question

Perform the indicated divisions. $$ \left(3 x^{4}+x^{3}-2 x^{2}-x+6\right) \div\left(x^{2}-1\right) $$

Step-by-Step Solution

Answer

The quotient is \(3x^2 + 1\).

Step 1: Set Up Synthetic Division

To perform the division \( \left(3x^4 + x^3 - 2x^2 - x + 6\right) \div \left(x^2 - 1\right) \), observe that the divisor \(x^2 - 1\) can be rewritten as \((x - 1)(x + 1)\). Therefore, we are dividing successively by \(x - 1\) and \(x + 1\). Let's start with \(x - 1\). List the coefficients of the dividend: 3, 1, -2, -1, and 6.

Step 2: Perform Synthetic Division by \(x - 1\)

Begin with the synthetic division using \(x - 1\). - Write down the root of the divisor, \(1\), on the left and the coefficients \(3, 1, -2, -1, 6\) on the right.- Bring down the 3.- Multiply 3 by 1, and write the result under the next coefficient. - Add the two numbers. Repeat for each coefficient.Performing these steps, the results yield new coefficients: 3, 4, 2, 1, and 7.

Step 3: Perform Synthetic Division by \(x + 1\)

Now using the quotient \(3, 4, 2, 1, 7\) from the first division, divide by \(x + 1\), which has a root of \(-1\).- Write down \(-1\) on the left.- Repeat the process of synthetic division starting with the quotient 3.The final coefficients after this division are: 3, 1, 0, 0. There is a remainder of 0, indicating that the division is exact.

Step 4: Write the Final Quotient

The sequence of coefficients obtained \(3, 1\) from the last division represents the polynomial \(3x^2 + 1\). This is the result of dividing \(3x^4 + x^3 - 2x^2 - x + 6\) by \(x^2 - 1\).

Key Concepts

Synthetic Division

Synthetic Division is a simplified method of dividing polynomials that can save time and reduce errors compared to the traditional long division approach. It is specifically useful when dividing a polynomial by a linear divisor, like \(x - c\). The main advantage of synthetic division is its ability to use only the coefficients of the polynomials, skipping the variable terms entirely.

• Start by identifying the root of the divisor, this can be found by setting the divisor equal to zero and solving for \x\. For example, for \(x - 1\), the root is \1\.
• List the coefficients of the dividend (the polynomial you are dividing) in order from the highest degree term to the constant.
• Perform the synthetic division process by writing the root to the left and carrying down the first coefficient.
• Multiply this carried-down number by the root and add this product to the next coefficient.
• Repeat the multiply-and-add process until you have processed all coefficients.

By using synthetic division twice in sequence, as done in this problem with roots 1 and -1, polynomial division becomes more manageable.

Quotient of Polynomials

The quotient of polynomials is the result you get when one polynomial is divided by another polynomial. In simpler terms, it's like performing division with numbers, but here we are dealing with expressions containing variables. The solution sought is typically in polynomial form itself and represents the whole number of times the divisor fits into the dividend.
To find the quotient:

• Perform the polynomial division (either through synthetic or long division) until all terms have been addressed.
• The coefficients derived from synthetic division steps can be used to write the quotient polynomial.
• Always keep in mind the degree must adjust accordingly; if dividing a degree 4 polynomial by a degree 2, the quotient should be a degree 2 polynomial.

In this case, after two rounds of division, the quotient obtained was \(3x^2 + 1\), illustrating that the original polynomial fits exactly \(x^2 - 1\) into it, producing no remainder.

Polynomial Roots

Polynomial roots, also known as zeros, are the values of \x\ for which the polynomial equals zero. Finding these roots is often a part of the polynomial division process as they are used within synthetic division to simplify calculations.

• The roots are solutions to the equation where the polynomial is set to zero. For example, for \(x - 1\), the root is \x = 1\.
• In synthetic division, these roots help in transforming the division process into a sequence of additive and multiplicative operations.
• Finding polynomial roots not only aids with division but also gives insights into the factorization and graphing of polynomials.

For the given division exercise, the roots \(1\) and \(-1\) were crucial in determining the quotient by systematically simplifying the division operations.

Polynomial Coefficients

Polynomial coefficients are the constants multiplied by each term in a polynomial expression. They indicate the "weight" or importance of each term in shaping the polynomial's graph and its properties. When performing synthetic division, it's only the coefficients that are needed, making the process efficient.

• The coefficients are written in descending order according to the polynomial's degree, from highest to constant.
• In our problem, the initial coefficients were 3, 1, -2, -1, and 6 representing \(3x^4 + x^3 - 2x^2 - x + 6\).
• Through synthetic division, these coefficients are methodically transformed, simplifying the polynomial step-by-step.
• After every round of synthetic division, the new set of coefficients represents the simplified polynomial or quotient.

Understanding how coefficients work and change during division can provide deeper insights into how polynomials behave under various operations.