Finding Zeros of Polynomial Functions
Polynomial Function:
Zeros:
Roots:
1. Find the zeros of the equation 𝑓(𝑥) = 2𝑥 2 + 7𝑥 − 15.
2. Find all the zeros of the equation 𝑓(𝑥) = 3𝑥 3 + 4𝑥 2 − 5𝑥 − 2.
3. Find all the zeros of the equation 𝑦 = 𝑥 3 + 𝑥 2 − 5𝑥 − 5.
4. Find all the real and complex zeros to 𝑥 4 − 𝑥 3 − 2𝑥 2 − 4𝑥 − 24 = 0.
5. Find all the roots of 𝑥 4 + 3𝑥 3 − 10𝑥 2 = 2𝑥 2 + 6𝑥 − 20.
Practice:
1. A line joining (0,10) and (5,0) and an inscribed
rectangle are shown. Find the coordinates (𝑥, 𝑦)
which maximizes the area of the rectangle.
4
2. Find the equation of the vertical parabola with
focus at (0,4), opening upward, and
containing the point (12,4).
(x,y)
2
5
10
3. Find the equation for the quadratic function 𝑓,
where 𝑓(−1) = 𝑓(−2) = 0 and 𝑓(0) = −4.
4. If 𝑓(𝑥) = |𝑥| + 1 and 𝑔(𝑥) = |𝑥| − 3, find the
zeros of g f .
5. Solve:
6. Working together, Robin and Lee can paint a
house in 4 days. Robin works twice as fast as
Lee. How long would it take each person
working alone to paint the house?
3
2
≥
𝑥+2 𝑥−4
Name:
CAH – 1/9/13: Finding Zeros
1. Find the equation of the vertical parabola that
contains the points (−6,0), (0,3), and (8,0).
2. If 𝑓 is a real valued function, find the sum of all
the distinct positive integers in the domain of 𝑓 if
𝑓(𝑥) = √6 − 𝑥.
3. Find the minimum value of the function 𝑓(𝑥) =
(𝑥 − 1)2 + 3
4. Solve:
5. Write an equation of the vertical parabola that
contains the points (−1,2) with focus (3, −1),
and which opens downward.
6. The arithmetic sequence with initial terms 60 and
180 has 48 terms. What is the 48th term?
7. Find the equation of the circle with center (2,1)
and tangent to the line 𝑥 − 3𝑦 = 9.
8. Solve:
4
− +3>0
𝑥
2𝑥 + 2𝑥+1 = 96
9. Find the equation of the parabola with focus
(−2, 1) and directrix 𝑥 = 6.
10. For what values of 𝑐 will the polynomial 𝑥 3 +
3𝑥 2 + 𝑐𝑥 − 6 have a zero at 𝑥 = 1?
11. Find the zeros of the equation:
𝑓(𝑥) = 2𝑥 3 + 9𝑥 2 − 20𝑥 − 75.
12. Simplify:
13. Find the equation of the parabola which is
symmetrical to the 𝑦-axis and passes through
the points (0, −2) and (2,0).
14. Let 𝑓(𝑥) = 𝑥 2 − 4𝑥 − 21. Find the values of 𝑥
such that 𝑓(|𝑥| − 8) = 0.
15. Solve:
16. Find the axis of symmetry and the range for the
quadratic function 𝑓(𝑥) = −3𝑥 2 − 6𝑥 + 8.
2𝑥 ≥ 16√2
6
(16𝑥)1/2 (𝑥 −2/3 )
𝑥 3/2