Questions: Find (if possible) the rational zeros of the function. (Enter your answers as a comma-separated list. If none exist, enter DNE.) h(t)=t^3+6t^2+9t+4

Solution Steps

To find the rational zeros of a polynomial function, we can use the Rational Root Theorem, which suggests that any rational solution, expressed as a fraction p/q, has p as a factor of the constant term and q as a factor of the leading coefficient. We then test these possible rational roots by substituting them into the polynomial to see if they yield zero.

To find the rational zeros of the polynomial \( h(t) = t^3 + 6t^2 + 9t + 4 \), we apply the Rational Root Theorem. This theorem states that any rational zero, expressed as \( \frac{p}{q} \), must have \( p \) as a factor of the constant term (4) and \( q \) as a factor of the leading coefficient (1).

The factors of the constant term 4 are \(\pm 1, \pm 2, \pm 4\). Since the leading coefficient is 1, the possible rational zeros are \(\pm 1, \pm 2, \pm 4\).

We test each possible rational zero by substituting it into the polynomial \( h(t) \) to see if it results in zero.

• \( h(-4) = (-4)^3 + 6(-4)^2 + 9(-4) + 4 = 0 \)
• \( h(-1) = (-1)^3 + 6(-1)^2 + 9(-1) + 4 = 0 \)

Both \(-4\) and \(-1\) are zeros of the polynomial.

Final Answer