Problem 11 - Intermediate Algebra

Question

If a polynomial is divided by \(3 a-2\) and the quotient is \(3 a^{2}+5\) with a remainder of \(6,\) how do we write the result?

Step-by-Step Solution

Answer

The polynomial is \(9a^3 - 6a^2 + 15a - 4\).

Step 1: Understand the Problem

We have a polynomial division problem. We know the divisor, which is \(3a - 2\), the quotient is \(3a^2 + 5\), and the remainder is \(6\). Our task is to express the polynomial in standard polynomial division form.

Step 2: Recall Polynomial Division Formula

In polynomial division, for a polynomial \(P(a)\), if it is divided by \(D(a)\) with a quotient \(Q(a)\) and a remainder \(R\), the polynomial can be expressed as: \(P(a) = D(a) \cdot Q(a) + R\).

Step 3: Substitute Known Values

We substitute the known values into the formula: \(D(a) = 3a - 2\), \(Q(a) = 3a^2 + 5\), and \(R = 6\). Thus, the polynomial \(P(a)\) is given as: \(P(a) = (3a - 2)(3a^2 + 5) + 6\).

Step 4: Expand the Expression

Expand the expression \((3a - 2)(3a^2 + 5)\). Multiply each term of \(3a - 2\) by each term of \(3a^2 + 5\):- \(3a \cdot 3a^2 = 9a^3\)- \(3a \cdot 5 = 15a\)- \(-2 \cdot 3a^2 = -6a^2\)- \(-2 \cdot 5 = -10\)Combine the terms: \(9a^3 - 6a^2 + 15a - 10\).

Step 5: Add the Remainder

Add the remainder to the expanded form: \(9a^3 - 6a^2 + 15a - 10 + 6\). Simplify by combining constants: \(9a^3 - 6a^2 + 15a - 4\).

Step 6: Conclusion

The polynomial \(P(a)\) is therefore expressed as \(9a^3 - 6a^2 + 15a - 4\) when divided by \(3a - 2\) with the given quotient and remainder.

Key Concepts

Remainder Theorem

The Remainder Theorem is a simple yet powerful tool in polynomial algebra. It helps to find the remainder when a polynomial is divided by a linear divisor of the form \(x - c\). When you have a polynomial \(P(x)\), and you know it is divided by \(x - c\), the Remainder Theorem states that the remainder of this division is simply \(P(c)\). This is because substituting \(c\) into the polynomial \(P(x)\) directly gives you the remainder.

For example, if you have the polynomial \(P(a)\) divided by \(3a - 2\), you can find the remainder by evaluating \(P(a)\) at \(a = \frac{2}{3}\). The result of this substitution gives you the value of the remainder without performing full polynomial division, thus speeding up calculations significantly, especially in complex polynomial functions.

Long Division of Polynomials

Long division of polynomials is similar to the long division we do with numbers. It is used to divide one polynomial by another, typically a lower-degree polynomial.

The steps in polynomial long division are:

• Divide the first term of the dividend by the first term of the divisor.
• Multiply the entire divisor by the result and subtract from the original dividend.
• Bring down the next term of the dividend and repeat the process.

This method helps us find both the quotient and the remainder, which can be assembled into the polynomial division format: \(P(a) = D(a) \cdot Q(a) + R\).

For example, dividing the polynomial \(9a^3 - 6a^2 + 15a - 4\) by \(3a - 2\) allows us to obtain this expression. Polynomial long division is a reliable method, especially when the divisor is not a simple binomial.

Synthetic Division

Synthetic division is a streamlined method for dividing a polynomial by a binomial of the form \(x - c\). It is a quick and less cumbersome alternative to long division, but it only applies in specific cases where the divisor is linear and the coefficients are integers or rational numbers.

To perform synthetic division:

• Write down the coefficients of the polynomial.
• Use the root of the divisor in the synthetic division setup. If you are dividing by \(x - c\), use \(c\).
• Bring down the first coefficient, multiply it by \(c\), and add it to the next coefficient, continuing this process for all coefficients.

The final number from this addition process gives you the remainder, while the other numbers form the coefficients of the quotient polynomial.

Synthetic division is especially useful for its speed and efficiency in reducing higher-order polynomials when conditions permit its use. However, it may not work if the divisor does not fit the required form, in which case long division is preferred.