1.) The chief designer for a large automobile company is considering 4 different radiator grills, 2 different style of headlights, and 5 different rear fender designs. If an automobile style consists of a grill, headlight, and rear fender, how many different styles of automobiles can be designed?

2.) Divide using long division.
(7x^3 + 16x^2 + 2x − 1) ÷ (x + 4)

3.) Divide using long division.
(3x^4 + 2x^3 − 8x + 6) ÷ (x^2 − 1)

4.) Write a polynomial in standard form with the given zeros. -3, -1, 2

#1

4*2*5

#2,#3: visit calc101.com and click on the long division button. It will show you all the details of polynomial division.

#4: (x+3)(x+1)(x-2)
Now just expand that. again, calc101.com has a button for polynomial multiplication.

The chief designer for a large automobile company is considering 4 different radiator grills, 2 different style of headlights, and 5 different rear fender designs. If an automobile style consists of a grill, headlight, and rear fender, how many different styles of automobiles can be designed?

1.) To find the number of different automobile styles that can be designed, we need to multiply the number of options for each component.

In this case, there are 4 radiator grill options, 2 headlight options, and 5 rear fender options.

To find the total number of combinations, we multiply the number of options for each component: 4 x 2 x 5 = 40.

Therefore, there are 40 different styles of automobiles that can be designed.

2.) To divide using long division, we start by dividing the first term of the dividend (7x^3) by the divisor (x + 4).

The result is 7x^2, which we write above the line. We then multiply this result by the divisor (x + 4) and subtract it from the dividend.

______________
x + 4 | 7x^3 + 16x^2 + 2x - 1
- (7x^3 + 28x^2)

This gives us -12x^2 + 2x - 1 as the new dividend. We then bring down the next term (-1x) from the original dividend.

_______________
x + 4 | 7x^3 + 16x^2 + 2x - 1
- (7x^3 + 28x^2)
_______________
-12x^2 + 2x - 1
- (-12x^2 - 48x)
_________________________
50x - 1

We repeat this process until we have no more terms to bring down. In this case, we have a remainder of 50x - 1.

Therefore, the result of the division is 7x^2 - 12x + 50 with a remainder of 50x - 1.

3.) To divide using long division, we start by dividing the first term of the dividend (3x^4) by the divisor (x^2 - 1).

The result is 3x^2, which we write above the line. We then multiply this result by the divisor (x^2 - 1) and subtract it from the dividend.

___________________
x^2 - 1 | 3x^4 + 2x^3 - 8x + 6
- (3x^4 - 3x^2)

This gives us 5x^3 - 5x^2 - 8x + 6 as the new dividend. We then bring down the next term (0x) from the original dividend.

_______________________
x^2 - 1 | 3x^4 + 2x^3 - 8x + 6
- (3x^4 - 3x^2)
_______________________
5x^3 - 5x^2 - 8x + 6
- (5x^3 - 5x^2)
_______________________
-3x^2 - 8x + 6

We repeat this process until we have no more terms to bring down. In this case, we have a remainder of -3x^2 - 8x + 6.

Therefore, the result of the division is 3x^2 + 5x - 3 with a remainder of -3x^2 - 8x + 6.

4.) To write a polynomial in standard form with the given zeros, we use the fact that if a number, say 'a', is a zero of the polynomial, then (x - a) is a factor of the polynomial.

The given zeros are -3, -1, and 2.

Therefore, the factors of the polynomial are (x - (-3)), (x - (-1)), and (x - 2), which simplify to (x + 3), (x + 1), and (x - 2).

To find the polynomial, we multiply these factors together:

(x + 3)(x + 1)(x - 2)

Expanding this expression:

(x + 3)(x + 1)(x - 2) = (x^2 + 4x + 3)(x - 2)
= x^2(x - 2) + 4x(x - 2) + 3(x - 2)
= x^3 - 2x^2 + 4x^2 - 8x + 3x - 6
= x^3 + 2x^2 - 5x - 6

Therefore, the polynomial in standard form with the given zeros is x^3 + 2x^2 - 5x - 6.