Problem 77 - A Graphical Approach to College Algebra

Question

Factor \(P(x)\) into linear factors given that \(k\) is a zero of \(P\). $$P(x)=x^{3}+5 x^{2}-3 x-15 ; \quad k=-5$$

Step-by-Step Solution

Answer

\((x + 5)(x - \sqrt{3})(x + \sqrt{3})\) are the linear factors.

Step 1: Verify that k is a zero

We need to verify that \( k = -5 \) is indeed a zero of \( P(x) \). Substitute \( x = -5 \) into \( P(x) \): \( P(-5) = (-5)^3 + 5(-5)^2 - 3(-5) - 15 \). This simplifies to \( -125 + 125 + 15 - 15 = 0 \). Therefore, \( k = -5 \) is a zero of \( P(x) \).

Step 2: Use Synthetic Division

Perform synthetic division on \( P(x) \) by the root \( x + 5 \). Write the coefficients: \( 1, 5, -3, -15 \). Use \( -5 \) as the divisor. Bring down the 1, multiply by -5, add to the next coefficient, repeat the process. The row computes to \( 1, 0, -3, 0 \), confirming zero remainder.

Step 3: Write the Polynomial as a Product

From synthetic division, we get the quotient polynomial \( x^2 - 3 \). Thus, the polynomial \( P(x) \) can be expressed as \((x + 5)(x^2 - 3)\).

Step 4: Factor the Quadratic Component

The quadratic \( x^2 - 3 \) can be factored further into \((x - \sqrt{3})(x + \sqrt{3})\). This is because \( x^2 - 3 = (x - \sqrt{3})(x + \sqrt{3}) \), where the roots are \( \sqrt{3} \) and \( -\sqrt{3} \).

Step 5: Combine All Linear Factors

Combine all the factors to express \( P(x) \) completely factored into linear terms. Thus, \( P(x) = (x + 5)(x - \sqrt{3})(x + \sqrt{3}) \).

Key Concepts

Synthetic Division

Synthetic Division is a shortcut method to divide a polynomial by a linear factor of the form \( x - k \). This is particularly effective when you know one root of the polynomial or wish to perform the division quickly.

Here's how you do it step by step:

• Write down the coefficients of the polynomial in descending order of power. For the polynomial \( P(x) = x^3 + 5x^2 - 3x - 15 \), these are 1, 5, -3, and -15.
• The divisor is the root \( k \) with its sign switched. So, if \( k = -5 \), you use 5.
• Bring down the leading coefficient. In this case, bring down the 1.
• Multiply this first coefficient by the divisor 5, and write the result under the next coefficient.
• Add the numbers in the column.
• Repeat the multiplication and addition steps for each column of coefficients. Insert the sums into the next column.

This process should be continued until every coefficient has been processed, and the last number gives you the remainder. If the remainder is zero, then the division is exact.

In our example, the computation yields a final form: \( 1, 0, -3, 0 \), confirming \(-5\) is indeed a root due to the zero remainder.

Roots of Polynomial

The concept of Roots of a Polynomial is central to factorization problems. A 'root' of a polynomial is a solution like \( x = k \) that makes \( P(x) = 0 \). Understanding roots helps us factorize the polynomial.

To determine whether a number \( k \) is a root, substitute \( k \) into the polynomial \( P(x) \). If resulting in zero, \( k \) is a root. This is precisely tested in our problem, where substituting \( k = -5 \) into \( P(x) \) yielded zero, confirming \(-5\) is a root.

Roots define polynomial behavior and help break down complex expressions into simpler parts. For instance, recognizing one root can often lead to discovering other potential factors and facilitate breaking the polynomial into linear or more manageable quadratic expressions. This step also sets the groundwork for synthetic division or further factorization.

Quadratic Factorization

Quadratic Factorization is pivotal when breaking down polynomials into more minor, linear factors. After synthetic division, polynomials often result in a quadratic expression, which might be factored further.

In our example, after performing synthetic division, we obtained \( x^2 - 3 \). To factor this, identify forms that can be broken down. Recognize that it is a difference of squares \( a^2 - b^2 = (a - b)(a + b) \).

Here, \( x^2 - 3 = (x - \sqrt{3})(x + \sqrt{3}) \). This form capitalizes on identifying perfect square terms—it's crucial to recognize patterns like these.

• Look for two terms, often a square of some variable \( x \) and a constant (for instance, 3 in this problem).
• Consider this constant as the square of some number, leading to terms like \( \sqrt{3} \).
• Express the quadratic as a product of two linear expressions as detailed above.

Quadratic factorization simplifies the polynomial into linear terms, completing the factorization process and making it easier to analyze or graph the polynomial, as was required in the original exercise to yield the final form \( (x + 5)(x - \sqrt{3})(x + \sqrt{3}) \).