Problem 12 - Precalculus : Building Concepts and Connections

Question

Show that the given value of \(x\) is a zero of the polynomial. Use the zero to completely factor the polynomial. $$p(x)=x^{3}-7 x+6 ; x=2$$

Step-by-Step Solution

Answer

When substituting \(x = 2\) in the polynomial \(x^{3} - 7x + 6\), it equals \(0\), therefore \(x = 2\) is indeed a zero of \(p(x)\). After performing polynomial division, the polynomial can be factored as \((x - 2)\) times the quotient from the division.

Step 1: Verify the Zero

Substitute \(x = 2\) into the given polynomial \(p(x) = x^{3} - 7x + 6\). If the result equals \(0\), then \(x = 2\) is indeed a zero of the polynomial: \(p(2) = 2^{3} - 7 * 2 + 6 = 0\)

Step 2: Divide Polynomial by $(x - 2)$

Next we use polynomial division to divide \(p(x) = x^{3}-7x+6\) by \((x - 2)\). This process should result in a quotient that is a binomial or lower degree polynomial.

Step 3: Write the Polynomial as a Product

As a result of the division, we can express the polynomial \(p(x)\) as a product of \((x - 2)\) and the quotient from the division in step 2. This process results in the factored form of the polynomial.

Key Concepts

Polynomial Division

Polynomial division is similar to long division with numbers, but instead we work with polynomials. When we divide one polynomial by another, our goal is to simplify the original polynomial into a simpler form.
Try to visualize rearranging terms so that they fit into a specific pattern. You can think of polynomial division as a process to break down complex polynomials.

• Start by determining the term that needs to be multiplied to the divisor to match the leading term of the dividend.
• Subtract the result from the original polynomial, then bring down the next term, just like in long division with numbers.
• Proceed until no terms are left.

In our exercise, the polynomial \(p(x) = x^{3} - 7x + 6\) is divided by \(x - 2\). This process helps us identify other factors of the polynomial, leading us to a factored form. Perform the division correctly to ensure the quotient reveals more about the polynomial's structure.

Zeros of Polynomials

Zeros of a polynomial are the values of \(x\) for which the polynomial equals zero. These are also referred to as roots or solutions of the polynomial equation. Finding zeros is crucial because they provide critical insights into the behavior and characteristics of the graph and the polynomial itself.
To verify whether a certain value is a zero of a polynomial, substitute the value into the polynomial and solve:

• If the result is zero, then that value is a zero of the polynomial.
• For example, plugging \(x = 2\) into \(p(x)\) confirmed that \(p(2) = 0\).

Identifying zeros is the first step towards completely factorizing a polynomial, leading us to uncover its simplest, most useful form. The zeros indicate where the graph of the polynomial touches or crosses the x-axis.

Factored Form of Polynomials

The factored form of a polynomial is a way of expressing the polynomial as a product of its factors. This form makes it easier to understand and solve polynomial functions, discover roots, and perform various analyses.
Once we identify a zero using polynomial division, we can express the polynomial as a product of its divisor and quotient. This results in a polynomial in its factored form, showing clearer insights into the polynomial's behavior.

• In our exercise, the polynomial \( p(x) = x^{3} - 7x + 6 \) is rewritten after confirming \(x = 2\) as a zero.
• It can be expressed as \((x - 2)\) multiplied by the quotient from the division.

The factored form is advantageous for solving equations, analyzing graphs, and integrating functions.