Question

Use synthetic division to divide. Divisor \(x-6\) Dividend $$3 x^{3}-16 x^{2}-72$$

Short answer

The quotient from the synthetic division of \(3x^{3} - 16x^{2} - 72\) by \(x - 6\) is \(3x^{2} + 2x + 12\).

Step-by-step solution

Arrange in descending order

Arrange both the divisor and the divisor in the standard form. In this case, both are already arranged properly. The divisor is \(x - 6\) and the dividend is \(3x^{3} - 16x^{2} - 72\).

Set up the Synthetic Division

Set up the synthetic division outline. Write the solution of \(x - 6 = 0\) (which is \(x = 6\)) on the left. Then write the coefficients of dividend \(3, -16, 0, -72\) to the right. The 0 is included for the missing \(x\) term in the dividend.

Perform Synthetic Division

Begin the synthetic division process. First bring down the first number of the dividend, which is 3. Then, the process is repeated: multiply the divisor by the number just written on the bottom row, place it in the next column, and then add down; keep doing this until every number in the top polynomial is gone. The calculations are as follows: multiply 6 (from the divisor) by 3 (last number in the bottom row), then place the product (18) to the bottom of -16 (the next number in the top row). Add them (18-16) to get 2. Repeat the process, multiply 6 by 2 to get 12, add to the 0 to get 12, and multiply 6 by 12 to get 72, add 72 and -72 to get 0. Therefore, the numbers in the bottom row are now 3, 2, 12, 0.

Write the quotient

The last step in synthetic division is to write down the quotient. Since we started with a third degree polynomial, we know the quotient will be degree 2. The coefficients from the division are 3, 2, and 12 which match the degree 2 polynomial, so they are the coefficients of the quotient polynomial. Our final quotient is \(3x^{2} + 2x + 12\), and the remainder is 0. Since the remainder is 0, \(x - 6\) is a factor of \(3x^{3} - 16x^{2} - 72\).

Key concepts

Polynomial Division

Polynomial division is an algebraic process similar to long division in arithmetic. It involves the division of one polynomial, called the dividend, by another polynomial, called the divisor, to find the quotient and possibly a remainder. The result expresses the original polynomial as the product of the divisor and the quotient, plus the remainder.

There are different methods to perform polynomial division, with synthetic division and algebraic long division being two commonly used techniques. Synthetic division is a shortcut method for when the divisor is a linear polynomial (of the form x - a), while algebraic long division works for divisors of any degree.

In our exercise, we are using synthetic division to divide a cubic polynomial by a linear factor, resulting in a quadratic polynomial as the quotient and determining if the linear factor is indeed a factor of the cubic polynomial.

Descartes' Rule of Signs

Descartes' Rule of Signs is a powerful theorem used to determine the number of positive and negative real roots of a polynomial equation. It is based on the observation of the number of sign changes in the coefficients of the terms when arranged in standard form.

For positive roots, the rule states that the number of positive real roots of a polynomial equation either equals the number of sign changes between consecutive coefficients or is less than that by an even number. On the other hand, to find the number of negative real roots, we first replace x with -x in the polynomial, and then apply the rule to this new polynomial.

While this rule gives us an insight into the possible number of real roots, it does not help identify the actual roots of the polynomial. However, it is an excellent preliminary tool before one embarks on root-finding techniques like synthetic division.

Algebraic Long Division

Algebraic long division is a method used when dividing polynomials, especially when the divisor is of higher degree (greater than one). This approach is much like the long division algorithm that many students learn in arithmetic, with similarities in the layout and steps - albeit applied to algebraic terms.

When performing algebraic long division, the dividend and divisor are arranged in long division format. The leading term of the divisor is divided into the leading term of the dividend, and the result is multiplied by the entire divisor. This product is then subtracted from the dividend, and the process is repeated with the remainder obtained until the remainder is either zero or of lower degree than the divisor.

The primary benefit of algebraic long division over synthetic division is that it can be used for divisors of any degree, making it a more flexible and comprehensive method.

Remainder Theorem

The Remainder Theorem is integral to the understanding of polynomial division, particularly when dealing with synthetic division or algebraic long division. It states that when a polynomial f(x) is divided by a linear divisor of the form x - a, the remainder of this division is the value of the polynomial f(a).

This means if the remainder is zero, the value f(a) is also zero, which implies that x = a is a root of the polynomial equation f(x) = 0. Therefore, the divisor x - a is a factor of the polynomial f(x).

Returning to our exercise, since the remainder after dividing the cubic polynomial by x - 6 is zero, by the Remainder Theorem, x = 6 is a root of the polynomial, and x - 6 is indeed a factor of the cubic polynomial 3x^3 - 16x^2 - 72.