Determine the x-intercepts?

Question

of.....
P(x)= (2x - 3)^7 (x+4)^8.

For each intercept determine whether the graph crosses the x-axis or intersects the x-axis without crossing it.

Answer 1

Ok.
In order to find the x intercepts, we want to see when P(x) = 0.
So, set:
0=(2x-3)^7 (x +4) ^8
We know that this will equal zero when either of the two factors (2x -3) or (x + 4) = 0
Thus:
2x-3 = 0
add 3 to both sides
2x =3
divide by 2
x = 3/2

x + 4 = 0
subtract 4 from both sides
x = -4

In order to tell whether or not P(x) crosses the axis at these two points, we want to set up a "sign pattern chart", describing the sign of P(x) (either positive or negative) at various points.

<----------------- -4 -------------------------------- 3/2 -------------------->

When x is to the left of -4:
(2x - 3) is negative, and (x +4) is also negative
(2x -3) ^7 is negative (a negative number raised to an odd power is negative) and ( x +4) ^ 8 is positive (anything to an even power(2,4,6,8...) is positive)
So we have P(x) = (-) * (+) = -
So, P(x) is less than zero (negative) on the left of -4
At x=-4, it equals 0
When x is to the right of -4 but less than 3/2
(2x -3) is still negative, but (x +4) is now positive
(2x -3)^7 is still negative and (x +4) ^8 is still positive
So we have P(x) = (-) * (+) = -
So, since P(x) is negative on the left of -4 and on the right of -4, we know P(x) intersects the x -axis without crossing it (a good way to describe this is that it "bounces" at x=-4)

Now, to the left of 3/2, we know that the sign of P(x) is negative(from what we just did)
At x=3/2, P(x) = 0
When x is to the right of 3/2 (x > 3/2)
(2x -3) is positive and (x +4) is positive.
Then (2x -3)^7 is positive and (x +4) ^8 is also positive (any positive number raised to any power is still positive)
So, for the sign of P(x) when x>3/2
P(x) = + * + = +
So, since P(x) is negative on the left of 3/2 and positive on the right of 3/2, we know that the graph crosses the x-axis at x = 3/2


Our sign pattern chart looks like this:

Sign of P(x)
P(x) is negative (-) P(x) is negative( -) P(x) is positive (+)

<------------------- -4 -------------------------------- 3/2 ----------------->

If P(x) changes sign (from - to + or from + to -) like it does at x = 3/2, it will cross the x-axis. If P(x) does not change signs ( like at -4), it will "bounce" off of the x-axis (not cross it).

Hope this helps!

Answer 2

P(x)= (2x - 3)^7 * (x+4)^8
P(0) gives y intercepts
P(0) = (2*0-3)^7 * (0+4)^8
p(0) = -2187*65536
p(0) = -143,327,232
(0, -143,327,232)

0 = (2x-3)^7 * (x+4)^8
x = -4, touches x-axis
x = 3/2, crosses x-axis

Answer 3

2x-3 = 0 --> x = 1.5
x+4 = 0 --> x = -4