Questions: For the polynomial function find the following: (i) Degree of the polynomial; (ii) All x intercepts; (iii) The y intercept. y=x^2-25

Transcript text: For the polynomial function find the following: (i) Degree of the polynomial; (ii) All $x$ intercepts; (iii) The $y$ intercept. \[ y=x^{2}-25 \]

Solution

Solution Steps

Solution Approach

Degree of the polynomial: The degree of a polynomial is the highest power of the variable $ x $. For the given polynomial $ y = x^2 - 25 $, the highest power of $ x $ is 2.
All $ x $ intercepts: The $ x $ intercepts are the values of $ x $ where $ y = 0 $. Set the polynomial equal to zero and solve for $ x $.
The $ y $ intercept: The $ y $ intercept is the value of $ y $ when $ x = 0 $. Substitute $ x = 0 $ into the polynomial and solve for $ y $.

Step 1: Degree of the Polynomial

The degree of a polynomial is the highest power of the variable $ x $. For the polynomial $ y = x^2 - 25 $, the highest power of $ x $ is 2.

Step 2: Finding the $ x $-Intercepts

The $ x $-intercepts are the values of $ x $ where $ y = 0 $. Set the polynomial equal to zero and solve for $ x $: \[ x^2 - 25 = 0 \] \[ x^2 = 25 \] \[ x = \pm 5 \] Thus, the $ x $-intercepts are $ x = -5 $ and $ x = 5 $.

Step 3: Finding the $ y $-Intercept

The $ y $-intercept is the value of $ y $ when $ x = 0 $. Substitute $ x = 0 $ into the polynomial: \[ y = 0^2 - 25 = -25 \] Thus, the $ y $-intercept is $ y = -25 $.