Questions: For the polynomial function find (i) the degree of the polynomial, (ii) all x-intercepts, and (iii) the y-intercept. y=x^2-196

Solution Steps

To solve the given polynomial function \( y = x^2 - 196 \), we need to determine the following:

(i) The degree of the polynomial: The degree is the highest power of \( x \) in the polynomial.

(ii) The x-intercepts: These are the values of \( x \) for which \( y = 0 \). Solve the equation \( x^2 - 196 = 0 \) to find the x-intercepts.

(iii) The y-intercept: This is the value of \( y \) when \( x = 0 \).

The polynomial function is given by \( y = x^2 - 196 \). The degree of the polynomial is determined by the highest power of \( x \) present in the expression. Here, the highest power is \( 2 \). Thus, the degree is \[ \text{Degree} = 2. \]

To find the x-intercepts, we set \( y = 0 \): \[ x^2 - 196 = 0. \] Solving for \( x \), we can factor the equation: \[ x^2 = 196 \implies x = \pm \sqrt{196} = \pm 14. \] Therefore, the x-intercepts are \[ x = -14 \quad \text{and} \quad x = 14. \]

The y-intercept occurs when \( x = 0 \): \[ y = 0^2 - 196 = -196. \] Thus, the y-intercept is \[ y = -196. \]

Final Answer