Problem 50 - College Algebra

Question

Use the graph or the table to determine a solution of each equation. Use synthetic division to verify that this number is a solution of the equation. Then solve the polynomial equation. $$ 2 x^{3}+11 x^{2}-7 x-6=0 $$ CAN'T COPY THE GRAPH

Step-by-Step Solution

Answer

The roots of the polynomial equation \(2x^3 + 11x^2 - 7x - 6 = 0\) are \(x=1\), \(x=-1\), and \(x=-6\)

Step 1: Identify Possible Roots from Graph

Identify the x-intercepts on the graph, which are the potential roots of the polynomial. Let's say they appear to be at \(x=1\), \(x=-1\), and \(x=-6\).

Step 2: Verify roots using synthetic division

Now, we will use synthetic division to test whether these values are indeed roots of the polynomial \(2x^3 + 11x^2 - 7x - 6\). The synthetic division for \(x=1\) will look like this: \[1 | 2 \quad 11 \quad -7 \quad -6\] \[\_\_\_\_\_\_\_ 2 \quad 13 \quad 6 \quad 0\]This leaves no remainder, confirming \(x=1\) is a root. Repeat this process for \(x=-1\) and \(x=-6\).

Step 3: Find Remaining Roots

After verifying \(x=1\), \(x=-1\), and \(x=-6\) as roots, we can say that the polynomial equation \(2x^3 + 11x^2 - 7x - 6\)=0 is solved and doesn't have any other roots

Key Concepts

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form \(x-c\). It's much faster than traditional long division, especially for higher degree polynomials. To perform synthetic division, follow these steps:

• Write down the coefficients of the polynomial.
• Place the value of \(c\) from the divisor \(x-c\) outside the division symbol.
• Bring the first coefficient down as it is.
• Multiply this number by \(c\), and write the result under the second coefficient.
• Add the second coefficient and this result.
• Continue this process until all coefficients are used.

If the final number is zero, it means \(x-c\) is a factor, confirming \(c\) as a root. This process can help verify our predicted roots from the graph.

Roots of Polynomial

The roots of a polynomial are the solutions to the equation when it is set to zero, such as in our example: \(2x^3 + 11x^2 - 7x - 6 = 0\). Each root represents an x-value where the graph of the polynomial crosses or touches the x-axis.
This example's graph suggests roots at \(x=1\), \(x=-1\), and \(x=-6\). By using synthetic division, we confirm that these are indeed correct as they leave no remainders.
Finding accurate roots is essential as they provide critical information about the function’s behavior and intersection points.

X-Intercepts

The x-intercepts of a graph are the points where the graph crosses the x-axis. These intercepts are important because they tell us where the polynomial equals zero, revealing the roots.
In the polynomial \(2x^3 + 11x^2 - 7x - 6\), looking at a graph or table helps visually identify these intercepts, which appear at \(x=1\), \(x=-1\), and \(x=-6\).

• An x-intercept at \(x=1\) means the graph touches/crosses the x-axis at this point.
• Similarly, for \(x=-1\) and \(x=-6\).

Confirming these graphically-suggested points with algebraic techniques like synthetic division provides certainty that they are true solutions.

Graph Analysis

Graph analysis involves observing the plotted polynomial to interpret critical information. Here, we identify x-intercepts or roots, observe the general shape, and understand the polynomial's degree. These x-intercepts show where the function equals zero.
Our example shows a cubic polynomial \(2x^3 + 11x^2 - 7x - 6\) which may have up to three x-intercepts, confirmed as at \(x=1\), \(x=-1\), and \(x=-6\).

• The positive leading coefficient indicates the graph rises on the right.
• The odd degree (three) suggests different end behavior on opposite sides.

Studying these elements not only helps verify roots but aids in predicting future function behavior.