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