Problem 14 - College Algebra with Corequisite Support

Question

For the following exercises, find the \(x\) - or t-intercepts of the polynomial functions. $$ f(x)=x^{3}+6 x^{2}-7 x $$

Step-by-Step Solution

Answer

The x-intercepts are x=0, x=-7, and x=1.

Step 1: Identify polynomial properties

Check the degree, leading coefficient, and whether it satisfies the definition of a polynomial (non-negative integer exponents only).

Step 2: Answer

The x-intercepts are x=0, x=-7, and x=1..

Key Concepts

Factoring Polynomials

Factoring polynomials is like breaking down a complicated expression into simpler parts, just like you would break down a sentence into words.
To factor a polynomial, we find its factors or numbers and expressions that can be multiplied together to get the original polynomial.

• First, identify common terms across the polynomial's components.
• In our example, each term in the polynomial \(f(x)=x^{3}+6x^{2}-7x\) includes \(x\), making it a common factor.
• By factoring \(x\) out of each term, we simplify our polynomial to \(x(x^2 + 6x - 7)\).

From here, further factorization may involve finding terms that, when multiplied, give the quadratic expression. Overall, factoring changes the polynomial into a form that is easier to work with, especially when finding intercepts.

Finding Intercepts

Finding the intercepts of a polynomial function is a way to determine where the graph of the function crosses the axes. For this exercise, we'll focus on the \(x\)-intercepts. Intercepts are points where the graph meets an axis. Understanding these will help in sketching the graph.

• The \(x\)-intercepts occur where the value of \(f(x)\) is zero. So, we solve \(f(x)=0\).
• In the case of \(f(x)=x(x^2 + 6x - 7)\), set it to zero: \(x(x^2 + 6x - 7) = 0\).
• Using the zero product property, which tells us if a product is zero, at least one factor must be zero. We solve \(x = 0\) and \(x^2 + 6x - 7 = 0\).

These solutions give us the \(x\)-intercepts. They are essential in understanding the polynomial's graph's shape.

Polynomial Functions

Polynomial functions are expressions involving terms summed together, where each term includes a variable raised to a power and multiplied by a coefficient. They are widely used in mathematics to model various phenomena.

• Consider the polynomial \(f(x) = x^3 + 6x^2 - 7x\), which is a third-degree polynomial because the highest power of \(x\) is three.
• Polynomial functions are continuous and smooth, meaning they do not have breaks or sharp angles.
• Understanding these functions helps in calculus, algebra, and real-world applications where modeling with precise behavior is crucial.

Polynomials also have properties such as end behavior and symmetry, which add to their shape and provide further information regarding their graphs. Knowing these key properties assists in predicting and understanding the nature of solutions and their graphs.