Determining Intercepts of Degree 4 Polynomial Function

TLDR; Determining vertical and horizontal intercepts of a degree 4 polynomial function by setting input or function value to zero and factoring the equation.

⭐ Vertical and Horizontal Intercepts

Given the polynomial function C of t, the task is to determine the vertical and horizontal intercepts.

To determine the horizontal intercepts, the function value is set equal to zero, while to determine the vertical intercept, the input variable is set equal to zero.

The intercepts are given as ordered pairs in the form of t comma C of t due to the function being of t.

The vertical intercept is found by setting t equal to zero, resulting in the ordered pair zero comma zero, which is the origin.

To find the horizontal intercepts, C of t is set to zero and solved for t.

📈 Determining Vertical Intercept

By setting t equal to zero, the function value C of zero is calculated, resulting in zero comma zero as the vertical intercept, which is the origin.

➗ Determining Horizontal Intercepts

C of t is set to zero, resulting in the equation zero equals two t to the fourth minus 10t cubed plus 12t squared.

The equation is solved by factoring, starting with factoring out the greatest common factor, which is two t squared, followed by factoring the trinomial into two binomials.

🔍 Factoring the Trinomial

The trinomial is factored into two binomials, resulting in factors of t minus three and t minus two.

Each of these factors is then set equal to zero and solved for t, giving the values of t where the function intersects the horizontal axis.

📊 Origin as Intercepts

The polynomial function passing through the origin results in it being both a horizontal and vertical intercept.

This is evident in the ordered pair zero comma zero being the origin and an intercept for both axes.

📈 Graphical Verification

The results are verified graphically by observing the graph of the polynomial function C of t.

The graph passing through the origin and having intercepts at two comma zero and three comma zero confirms the correctness of the determined intercepts.