Problem 78 - A Graphical Approach to College Algebra

Question

Factor \(P(x)\) into linear factors given that \(k\) is a zero of \(P\). $$P(x)=x^{3}+9 x^{2}-7 x-63 ; \quad k=-9$$

Step-by-Step Solution

Answer

\(P(x) = (x + 9)(x - \sqrt{7})(x + \sqrt{7})\)

Step 1: Use the Zero to Create a Factor

Since it is given that \(k = -9\) is a zero of \(P(x)\), \(x + 9\) is a factor of \(P(x)\).

Step 2: Perform Synthetic Division

Use synthetic division to divide \(P(x)\) by \(x + 9\). Place \(-9\) outside the division box and write the coefficients of \(P(x)\) — which are \(1, 9, -7, -63\) — inside. Carry out the synthetic division process:1. Bring down the leading coefficient: \(1\).2. Multiply \(-9\) by \(1\) and add to \(9\) to get \(0\).3. Multiply \(-9\) by \(0\) and add to \(-7\) to get \(-7\).4. Multiply \(-9\) by \(-7\) and add to \(-63\) to get \(0\).The bottom row gives the quotient: \(1, 0, -7\), so the reduced polynomial is \(x^2 - 7\). The final \(0\) confirms that \(x + 9\) is a factor.

Step 3: Factor the Quotient Polynomial

The polynomial from the division is \(x^2 - 7\). It cannot be factored further over the real numbers because it does not have real roots. However, it can be expressed using complex factors as \((x - \sqrt{7})(x + \sqrt{7})\).

Step 4: Write the Complete Factorization

Combine the factors from Step 1 and Step 3:\[ P(x) = (x + 9)(x - \sqrt{7})(x + \sqrt{7}). \]

Step 5: Verify the Factorization

Multiply the factors back together to verify that they yield the original polynomial. Verify the multiplication:- \((x - \sqrt{7})(x + \sqrt{7}) = x^2 - 7\).- Then multiply by \((x + 9)\): \((x + 9)(x^2 - 7) = x^3 + 9x^2 - 7x - 63\), which matches \(P(x)\).

Key Concepts

Synthetic Division

Synthetic division is a streamlined version of polynomial division, particularly suited for dividing by linear factors of the form \(x - k\). It uses less writing and makes handling of subtraction much easier. Instead of dividing traditional polynomial long division, synthetic division simplifies the process and is quicker.
To perform synthetic division, you:

• Identify the zero, \(k\), that you are using to divide, which comes from your factor \(x - k\).
• Write the coefficients of the polynomial in order.
• Bring the first coefficient straight down.
• Multiply \(k\) by the value just brought down and add it to the next coefficient.
• Continue this process of multiplication and addition until all coefficients are used up.

The last number you get, if zero, confirms that \(x - k\) is a factor of the polynomial. This process provides the coefficients of the quotient polynomial.

Complex Numbers

Complex numbers are an extension of the real numbers and include imaginations, that is, they have a real part and an imaginary part. The imaginary unit is represented as \(i\), where \(i = \sqrt{-1}\). Complex numbers are written as \(a + bi\).
In the context of polynomial factorization, sometimes the polynomial does not have real factors, as shown in the exercise where \(x^2 - 7\) cannot be factored further in the real number plane.

• Instead, we move to the complex plane, using imaginary numbers.
• For example, \(x^2 - 7\) can be expressed as \((x - \sqrt{7})(x + \sqrt{7})\) because \(\sqrt{7}\) can be interpreted as a complex number with an imaginary component if needed.

This allows for polynomial factorization even when real roots do not exist.

Zero of a Polynomial

A zero of a polynomial is a value of \(x\) that makes the polynomial equal to zero. This value means that \(x - k\) is a factor of the polynomial. Knowing a zero is crucial for factoring the polynomial.
Here's how to use it:

• Find a zero of the polynomial, provided as \(k\).
• Use this zero to derive a factor \((x - k)\) for the polynomial.
• Perform division using this factor to simplify the polynomial.

This method is extremely efficient especially in higher degree polynomials where multiple zeros provide multiple factors. In the exercise example, \(-9\) is clearly stated as a zero, leading to the factorization of \(x + 9\). It becomes a building block for reducing and further analyzing the polynomial.