3.3, (13) Consider the following

Question

P(x) = x³ + 2x² − 9x − 18,    c = −2

Show that the given value of c is a zero of P(x).

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Find all other zeros of P(x). (Enter your answers as a comma-separated list. Enter all answers including repetitions.)

x = 

Answer 1

Hi Tori,

To indicate that -2 is a zero, just plug -2 in for x:

P(-2) = (-2)³ + 2(-2)² -9(-2) -18 = 0

Now, for the other zeros, factor by grouping:

x³ + 2x² - 9x - 18 = 0

x² (x + 2) -9 (x + 2)

This simplifies to:

(x² - 9) (x+2)

You already know x= -2 is a zero. To find the others, recall that x² - 9 is a difference of two squares, which factors to:

(x+3) (x-3)

Thus, the other two zeros are:

x = 3

x = -3

I hope this helps.

Answer 2

The question doesn't mention to find it in a certain way. Graph the equation on Desmos. You'll see the zeroes are -3, 2, 3.

Answer 3

We can confirm that -2 is a zero by synthetic division and confirming the remainder is zero

-2 | 1 2 -9 -18

-2 0 18

1 0 -9 0 <- There is the zero remainder to confirm -2 is a zero

To find the other zeros, take what is left from synthetic division

x²-9=0 Factor

(x-3)(x+3)=0

The remaining zeros are -3,3