91Ó°ÊÓ

Factoring polynomials helps simplify complex polynomial expressions into products of simpler polynomials. This step is crucial when solving polynomial equations or graphing polynomial functions.

In our exercise, we start with the polynomial function:\[ f(x) = 2x^4 + x^3 - 6x^2 - 7x - 2 \]
Factoring this polynomial involves finding roots that satisfy the polynomial, then breaking it down into simpler factors. We used the Rational Root Theorem to find potential roots and tested these using substitution.

Once a root is found (e.g., \( x = -1 \)), we perform polynomial division to simplify the polynomial further. In this exercise, we factor \( f(x) \) completely to:\[ f(x) = (x + 1)(2x + 1)(x^2 - 2) \]

The Rational Root Theorem provides a list of possible rational roots for a polynomial equation. This theorem states that any rational solution \( \frac{p}{q} \) of the polynomial equation \( a_n x^n + a_{n-1} x^{n-1} + \.\.\.+ a_0 = 0 \) (with \( a_n \) and \( a_0 \) non-zero) must satisfy:

• \( p \) (numerator) divides the constant term \( a_0 \)
• \( q \) (denominator) divides the leading coefficient \( a_n \)

For our polynomial \( 2x^4 + x^3 - 6x^2 - 7x - 2 \), the possible rational roots are ±1, ±2.

We test these values to determine which ones are actual roots by substituting them into the polynomial. If the result is zero, it confirms that the value is a root.

Polynomial division is a technique used to divide one polynomial by another, resulting in a quotient and possibly a remainder. This method simplifies the higher-degree polynomial into smaller parts that are easier to manage.

In our solution, after finding a root (e.g., \( x + 1 \)), we divide the original polynomial \[ f(x) = 2x^4 + x^3 - 6x^2 - 7x - 2 \] by \( x + 1 \) using polynomial or synthetic division. The quotient here is \( 2x^3 - x^2 - 7x - 2 \).

This process continues until all factors of the polynomial are found, which ultimately simplifies the original polynomial into a product of lower-degree polynomials.

Graphing polynomials involves plotting the polynomial function on a coordinate plane. Knowing the factored form helps identify the roots (x-intercepts) and analyze the end behavior of the function.

For the factored form of our polynomial \[ f(x) = (x + 1)(2x + 1)(x^2 - 2) \]
We use these steps:

• Find x-intercepts: Set \( f(x) = 0 \), solve for \( x \). The roots are \( x = -1, -\frac{1}{2}, \sqrt{2}, -\sqrt{2} \)
• Determine end behavior: Since the leading term in the expanded form is \( 2x^4 \), the graph rises on the left and on the right.
• Plot additional points: Choose x-values, substitute in \( f(x) \), and plot the points.
• Sketch graph: Connect the points using smooth curves considering the end behavior and intercepts.

By following these steps, we obtain a clear visual representation of the polynomial function.