HELP IN PRECALC HOMEWORK?

Question

Update : Consider the following.
P(t) = t4 − 18t2 + 32
(a) Use the zero or root feature of a graphing utility to approximate the zeros of the function accurate to three decimal places. (Enter your answers as a comma-separated list.)
t =

(b) Determine the exact value of one of the zeros (use synthetic division to verify your result).
t =

(c) Factor the polynomial completely.
P(t) =

Answer 1

a) The polynomial function P(t) = t⁴ - 18t² + 32

Test points

Make the table of values to for the polynomial.

Choose random values for t and find the corresponding values for P(t).

t	P(t) = t4 - 18t2 + 32	(t, P(t))
- 3	P(- 3) = (- 3)4 - 18(- 3)2 + 32	(-3, -49)
-2	P(- 2) = (- 2)4 - 18(- 2)2 + 32	(- 2, - 24)
0	P(0) = (0)4 - 18(0)2 + 32	(0,32)
1	P(1) = (1)4 - 18(1)2 + 32	(1, 15)
2	P(2) = (2)4 - 18( 2)2 + 32	(2, - 24)
3	P(3) = (3)4 - 18(3)2 + 32	(3, - 49)

End behavior P(t) = t⁴ - 18t² + 32

Degree of polynomial is 4 and leading coefficient 1.

The graph of a polynomial function is always a smooth curve; that is, it has no breaks or corners.

All even degree polynomials behave on their ends like quadratics.

All even degree polynomials are either up on both ends and or down on both ends.depending on whether the polynomial has, respectively, a positive or negative leading coefficient.

The above polynomial even degree  polynomial with a positive leading coefficient .

So the graph up on both ends.

Graph

1.Draw a coordinate plane.

2.Plot the coordinate points found in the table.

3.Then sketch the graph, connecting the points with a smooth curve.

The points where it crosses the t axis  will give solutions to the polynomial function P(t) .

From the graph the zeros are t = {-4, -1.414, 1.414, 4}.

Answer 2

c) P(t) = t⁴ - 18t² + 32

= t⁴ - 2t² - 16t² + 32

= t²(t² - 2) - 16(t² - 2)

= (t² - 2)(t² - 16)

= [t² - (√2)²][t² - 4²]

Apply the formula (a² - b²) = (a - b)(a + b)

= (t + √2)(t - √2)(t + 4)(t - 4)

Factoring of P(t) = t⁴ - 18t² + 32 = (t + √2)(t - √2)(t + 4)(t - 4).

Answer 3

b) P(t) = t⁴ - 18t² + 32

= t⁴ - 2t² - 16t² + 32

= t²(t² - 2) - 16(t² - 2)

= (t² - 2)(t² - 16)

= [t² - (√2)²][t² - 4²]

Apply the formula (a² - b²) = (a - b)(a + b)

= (t + √2)(t - √2)(t + 4)(t - 4)

From Factor theorem,

When x - c is a factor of the polynomial then f(c) = 0

So - √2, √2, -4 and 4 are real zeros of P(t).

Write the coefficients as shown below.

Here we check the one of the real zero -4.

The remainder is zero.

Therefore, - 4 is one of the zero of the P(t).