What is a cubic polynomial function in standard form with zeros 1, –2, and 2?

Use synthetic division to find P(3) for P(x) = x^4 – 6x^3 – 4x^2 – 6x – 2.

Divide 3x^3 + 3x^2 + 2x – 2 by x + 3 using long division.

Divide using synthetic division.
–3x3 + 11x2 – x + 20 by (x – 4).

p(x) = (x-1)(x+2)(x-2) = x^3 - x^2 -4x + 4

P(3) = -137
See http://easycalculation.com/algebra/synthetic-division-calculator.php

visit calc101.com and click on the long division link

synthetic division p(x)x^3-5x^2+10

To find a cubic polynomial function in standard form with zeros 1, -2, and 2, we can use the fact that if a number "r" is a zero of a polynomial function, then (x - r) is a factor of the polynomial.

Let's start by finding the factors for the given zeros:
(x - 1), (x + 2), and (x - 2)

Now we can write the cubic polynomial function in standard form by multiplying these factors:

P(x) = (x - 1)(x + 2)(x - 2)
= (x^2 + 2x - x - 2)(x - 2)
= (x^2 + x - 2)(x - 2)
= x^3 + x^2 - 2x - 2x^2 - 2x + 4
= x^3 - x^2 - 4x + 4

So, the cubic polynomial function in standard form with zeros 1, -2, and 2 is P(x) = x^3 - x^2 - 4x + 4.

Now, let's find P(3) using synthetic division for the polynomial P(x) = x^4 – 6x^3 – 4x^2 – 6x – 2.

Using synthetic division, we write the coefficients of the polynomial (including the zero placeholders) as follows:

3 | 1 -6 -4 -6 -2
| 3 -9 -39
_______________________
1 -3 -13 -45

The remainder after dividing is -45. Therefore, P(3) = -45.

Next, let's divide 3x^3 + 3x^2 + 2x - 2 by x + 3 using long division.

3x^2 + ( -6x + 20 )
____________________________
x + 3 | 3x^3 + 3x^2 + 2x - 2
- (3x^3 + 9x^2)
_______________________
-6x^2 + 2x - 2
- (-6x^2 - 18x)
_______________________
20x - 2
- (20x + 60)
___________________________
- 62

The remainder after dividing is -62.