Lesson 7 Building Strong Roots Solidify Understanding

Ready

Divide using long division. (These problems have no remainders. If you get one, try again.)

1.

$x + 3 5 x^{3} + 2 x^{2} - 45 x - 18$

2.

If $p (x) = 5 x^{3} + 2 x^{2} - 45 x - 18$ , what is the value of $p (- 3)$ ?

3.

$x - 6 x^{3} - x^{2} - 44 x + 84$

4.

If $p (x) = x^{3} - x^{2} - 44 x + 84$ , what is the value of $p (6)$ ?

5.

$x - 5 3 x^{3} - 15 x^{2} + 12 x - 60$

6.

If $p (x) = 3 x^{3} - 15 x^{2} + 12 x - 60$ , what is the value of $p (5) ?$

7.

$x + 2 x^{4} + 6 x^{3} + 7 x^{2} - 6 x - 8$

8.

If $p (x) = x^{4} + 6 x^{3} + 7 x^{2} - 6 x - 8$ , what is the value of $p (- 2)$ ?

Set

Predict the number of roots for each of the given polynomial equations.

9.

$a (x) = x^{2} + 3 x - 10$

10.

$b (x) = x^{3} + x^{2} - 9 x - 9$

11.

$c (x) = - 2 x - 4$

12.

$d (x) = x^{4} - x^{3} - 4 x^{2} + 4 x$

13.

$f (x) = - x^{2} + 6 x - 9$

14.

$g (x) = x^{6} - 5 x^{4} + 4 x^{2}$

The graphs of the polynomials from the previous problems are shown. Check your predictions. Then, use the graph to help you write the polynomial in factored form.

15.

$a (x) = x^{2} + 3 x - 10$

16.

$b (x) = x^{3} + x^{2} - 9 x - 9$

17.

$c (x) = - 2 x - 4$

18.

$d (x) = x^{4} - x^{3} - 4 x^{2} + 4 x$

19.

$f (x) = - x^{2} + 6 x - 9$

20.

$g (x) = x^{6} - 5 x^{4} + 4 x^{2}$

21.

The graphs of problems 19 and 20 don’t seem to follow the Fundamental Theorem of Algebra, but there is something similar about each of the graphs. Explain what is happening at the point $(3, 0)$ in problem 19 and at the point $(0, 0)$ in problem 20.

Go

Find the roots of each quadratic function.

22.

$f (x) = x^{2} + 20 x + 51$

23.

$f (x) = x^{2} + 10 x + 25$

24.

$f (x) = 3 x^{2} + 12 x$

25.

$f (x) = x^{2} - 11$

26.

$f (x) = x^{2} + x - 1$

27.

$f (x) = x^{2} + 2 x + 3$