Problem 79 - A Graphical Approach to College Algebra

Question

Factor \(P(x)\) into linear factors given that \(k\) is a zero of \(P\). $$P(x)=x^{3}-2 x^{2}-7 x-4 ; \quad k=-1$$

Step-by-Step Solution

Answer

The linear factors of \( P(x) \) are \((x + 1)^2(x - 4)\).

Step 1: Confirm k as a Zero

To confirm that \( k = -1 \) is a zero of the polynomial \( P(x) = x^3 - 2x^2 - 7x - 4 \), substitute \( x = -1 \) into \( P(x) \) and check if the result is 0.\[ P(-1) = (-1)^3 - 2(-1)^2 - 7(-1) - 4 = -1 - 2 + 7 - 4 = 0. \]Since \( P(-1) = 0 \), \( k = -1 \) is indeed a zero of \( P(x) \).

Step 2: Perform Synthetic Division

Use synthetic division to divide \( P(x) \) by \( (x + 1) \), since \( x = -1 \) is a root. Write down the coefficients of \( P(x) \): \[ 1, -2, -7, -4. \]Perform synthetic division:\[\begin{array}{ccc|c} & -1 & \ \hline 1 & -2 & -7 & -4 \ & & -1 & 3 & 4 \ \hline & 1 & -3 & -4 & 0 \ \end{array}\]The quotient is \( x^2 - 3x - 4 \) and there is no remainder.

Step 3: Factor the Quotient

Factor the quadratic \( x^2 - 3x - 4 \). Look for two numbers that multiply to \(-4\) and add to \(-3\): they are \(-4\) and \(1\). Thus, \( x^2 - 3x - 4 \) can be factored as \((x - 4)(x + 1)\).

Step 4: Combine All Linear Factors

The original polynomial \( P(x) = x^3 - 2x^2 - 7x - 4 \) can be expressed as the product of its linear factors, given that \( k = -1 \) is a zero and the quotient was factored. Therefore, the complete factorization is: \[ P(x) = (x + 1)(x - 4)(x + 1). \]

Step 5: Simplify the Factors

Since \( (x + 1) \) appeared twice, we can express the polynomial more succinctly as: \[ P(x) = (x + 1)^2(x - 4). \]This is the final factorization of \( P(x) \) into its linear factors.

Key Concepts

Synthetic Division

Synthetic division is a simplified method of dividing a polynomial by a linear factor of the form \(x - k\). It is a quicker and more efficient alternative to the long division method.

This method uses only the coefficients of the polynomial, making it less cumbersome. It's particularly useful when you know that a certain value is a zero of the polynomial.

Here's a simple way to think about synthetic division:

• Write down the coefficients of the polynomial in descending order of the powers.
• Next to the coefficients, write the zero, \(k\), of the divisor \((x - k)\).
• Bring down the first coefficient unchanged.
• Multiply this value by \(k\) and add it to the next coefficient.
• Repeat this process until you finish with all coefficients.
• The numbers you get at the bottom, except for the last one, represent the coefficients of the quotient polynomial. The last number is the remainder.

In the exercise, using synthetic division with \(k = -1\) confirms that \(x + 1\) is a factor of \(P(x)\), and it helps you find the quotient \(x^2 - 3x - 4\).

Zero of a Polynomial

A zero of a polynomial is a value that, when substituted for the variable, results in the zero value for the polynomial. In other words, if \(k\) is a zero of the polynomial \(P(x)\), then \(P(k) = 0\). Identifying zeros of polynomials is crucial because it leads to their factorization.

When you have a polynomial like \(P(x) = x^3 - 2x^2 - 7x - 4\), knowing that \(k = -1\) is a zero helps you to break down the polynomial into simpler components. You confirm this by substituting \(x = -1\) into the polynomial and checking if it equals zero, just as the exercise shows.

• If \(P(k) = 0\), \(x - k\) is a factor of the polynomial.
• Zeros reveal the roots of the polynomial equation \(P(x) = 0\).
• These roots are where the graph of the polynomial crosses or touches the x-axis.

In our example, since \(-1\) is a zero, the factor is \((x + 1)\), which can be used in the synthetic division process.

Quadratic Factoring

Once a polynomial is reduced using synthetic division, it often results in a simpler quadratic. Factoring quadratics is generally more straightforward and is a critical step toward polynomial factorization.

Consider the quadratic \(x^2 - 3x - 4\). Factoring it involves finding two numbers that multiply to the constant term (-4) and add up to the linear coefficient (-3).

Here is a simple strategy to factor quadratics:

• List pairs of numbers that multiply to the constant term.
• Identify which pair adds up to the linear term.
• Rewrite the middle term using your two numbers.
• Factor by grouping.

In the quadrant \(x^2 - 3x - 4\), the numbers -4 and 1 multiply to -4 and add to -3, leading to the factors \((x - 4)(x + 1)\).

Quadratic factoring completes the polynomial factorization, helping originally complex problems transform into their simplest linear factors before solving.