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Given the function $P(t)=2 t^4+26 t^3+84 t^2$:

its $P$-intercept is □

its $t$-intercepts are □

Verified Answer :

  • The P-intercept is found by evaluating \(P(0)\), which is \(0\).
  • The t-intercepts are found by solving the equation \(2t^4 + 26t^3 + 84t^2 = 0\).
  • Factoring the equation as \(2t^2(t+6)(t+7) = 0\), the intercepts are \(t = 0, -6, -7\).
  • The P-intercept is \(\boxed{0}\) and the t-intercepts are \(\boxed{0, -6, -7}\).

Explanation

  1. Finding the P-intercept

To find the P-intercept of the function \(P(t) = 2t^4 + 26t^3 + 84t^2\), we evaluate the function at \(t = 0\):
\[P(0) = 2(0)^4 + 26(0)^3 + 84(0)^2 = 0 + 0 + 0 = 0.\]
Thus, the P-intercept is \(0\).

  1. Finding the t-intercepts

To find the t-intercepts of the function \(P(t) = 2t^4 + 26t^3 + 84t^2\), we set the function equal to zero and solve for \(t\):
\[2t^4 + 26t^3 + 84t^2 = 0.\]
We can factor out the common term \(2t^2\):
\[2t^2(t^2 + 13t + 42) = 0.\]
Next, we factor the quadratic expression \(t^2 + 13t + 42\). We look for two numbers that multiply to \(42\) and add to \(13\). These numbers are \(6\) and \(7\). So, the expression becomes:
\[2t^2(t + 6)(t + 7) = 0.\]
Setting each factor to zero gives the roots:

  1. \(2t^2 = 0 \implies t = 0\)
  2. \(t + 6 = 0 \implies t = -6\)
  3. \(t + 7 = 0 \implies t = -7\)

Thus, the t-intercepts are \(0, -6,\) and \(-7\).

Examples

In physics, a projectile's height \(P\) over time \(t\) can often be modeled by a polynomial function. The P-intercept represents the initial height of the object at \(t=0\), while the t-intercepts represent the times when the object is at ground level (\(P=0\)). Knowing these points helps in determining the duration of the flight or the starting point of the motion.

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