Problem 21 - A Graphical Approach to College Algebra

Question

Find each quotient when \(P(x)\) is divided by the binomial following it. $$P(x)=x^{3}+2 x^{2}-17 x-10 ; \quad x+5$$

Step-by-Step Solution

Answer

Quotient is \(x^2 - 3x - 2\); remainder is 0.

Step 1: Set Up Synthetic Division

To begin, set up synthetic division using the zero of the divisor. The divisor is \(x + 5\), so the zero is \(-5\). Write down \(-5\) on the left and list the coefficients of the polynomial \(P(x) = x^3 + 2x^2 - 17x - 10\), which are \(1, 2, -17, -10\).

Step 2: Bring Down the Leading Coefficient

Bring down the first coefficient, \(1\), directly to the bottom row.

Step 3: Multiply and Add

Multiply the zero \(-5\) by the number just written \(1\) and write the result \(-5\) under the next coefficient \(2\). Add \(2\) and \(-5\) to get \(-3\), and write \(-3\) below the line.

Step 4: Repeat Multiply and Add

Repeat the same process: Multiply \(-5\) by \(-3\) to get \(15\), write it under the \(-17\), and add to get \(-2\). Then multiply \(-5\) by \(-2\) to get \(10\), write it under the \(-10\), and add to get \(0\).

Step 5: Write the Quotient

The result row \(1, -3, -2, 0\) represents the coefficients of the quotient polynomial. Since we started with a degree 3 polynomial, the quotient will be one degree less: \(1x^2 - 3x - 2\). The remainder is \(0\), indicating \(x + 5\) is a factor.

Key Concepts

Polynomial Division

Polynomial division is a method of dividing a polynomial by another polynomial. It's similar to long division with numbers, but instead of digits, you work with the coefficients of the polynomial terms. In this technique, you rewrite the polynomial as a division problem to find the quotient, which is a simpler polynomial.When dealing with polynomial division, you typically use either long division or synthetic division.

• **Long Division**: Similar to numerical long division, it's a straightforward but sometimes lengthy process. You subtract polynomials at each step to solve for the quotient.
• **Synthetic Division**: A more efficient method, but it's limited to dividing by linear divisors of the form \(x + c\).

In our exercise, we use synthetic division. Once completed, you find the quotient and check if there's a remainder. If the remainder is zero, as it was here, the divisor perfectly divides the original polynomial.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors. This process is crucial in simplifying polynomial expressions and solving polynomial equations. It essentially breaks down a complex polynomial into simpler components, making it easier to work with.The key steps in factoring polynomials include:

• Finding common factors in each term, which you can factor out.
• Identifying special patterns such as the difference of squares or perfect square trinomials.
• Using techniques like grouping for polynomials with four or more terms.

In the context of our example:- Since the remainder was zero in the division, the binomial \(x + 5\) is a factor of the polynomial \(P(x)\).- This is a common scenario in polynomial division, which helps identify roots of polynomials by showing if the divisor is a factor.

Binomial Division

Binomial division is a specific type of polynomial division where a polynomial is divided by a binomial, typically of the form \(x - c\). The solution helps determine how many times the divisor fits into the polynomial, revealing factors and roots.Synthetic division is often used for quick and effective binomial division:

• The divisor is rewritten or adjusted to find its root, \(x + c\), which is \(-c\).
• This value, which acts as the zero of the binomial, is used in synthetic division to simplify the process.

In our example:- The divisor \(x + 5\) gives a zero of \(-5\).- By using \(-5\) in synthetic division, we determined that \(x + 5\) divides the polynomial evenly (since the remainder was zero).Overall, binomial division, particularly when paired with synthetic division, can expedite calculations and clarify polynomial structures efficiently.