David paints 3 rooms in 7.5 hours at the same pace how long would it take him to paint 15 rooms

Factor the cubic polynomial 6x3 – 11x2 – 12x 5. use the rational root theorem and synthetic division

pochemuha

The factorization of given cubic polynomial 6x³ - 11x² - 12x + 5 is:

6x³ - 11x² - 12x + 5 = (x + 1)(x - $\frac{5}{2}$ )(x - $\frac{1}{3}$ )

For given question,

We have been given the cubic polynomial 6x³ - 11x² - 12x + 5

We need to factorize given cubic polynomial.

By the rational roots theorem, any rational zero of f(x) is expressible in the form ± $\frac{p}{q}$ for integers p, q with p a divisor of the constant term 5 and q a divisor of the coefficient 6 of the leading term.

Factors of p = 5: 1, 5

Factors of q = 6: 1, 2, 3, 6

That means that the only possible rational zeros are:

±{ $\frac{1}{1} ,\frac{1}{2} ,\frac{1}{3} ,\frac{1}{6} ,\frac{5}{1} ,\frac{5}{2} ,\frac{5}{3} ,\frac{5}{6}$ }

= ±{ $1 ,\frac{1}{2} ,\frac{1}{3} ,\frac{1}{6} ,5 ,\frac{5}{2} ,\frac{5}{3} ,\frac{5}{6}$ }

We need to find the exact zeros of given cubic polynomial.

For x = 1,

6(1)³ - 11(1)² - 12(1) + 5 = -12

This means, x = 1 is not a zero of given cubic polynomial.

For x = -1,

6(-1)³ - 11(-1)² - 12(-1) + 5 = 0

This means, x = -1 is a zero of given cubic polynomial and (x + 1) is a factor.

To factorize given cubic polynomial we use synthetic division.

The synthetic division (6x³ - 11x² - 12x + 5) ÷ (x + 1) is as shown in following image.

⇒ 6x³ - 11x² - 12x + 5 = (x + 1)(6x² - 17x + 5)

The factors of above quadratic polynomial 6x² - 17x + 5 are:

⇒ 6x² - 17x + 5 = (x - $\frac{5}{2}$ )(x - $\frac{1}{3}$ )

So, the factors of given cubic polynomial are:

⇒ 6x³ - 11x² - 12x + 5 = (x + 1)(x - $\frac{5}{2}$ )(x - $\frac{1}{3}$ )

Therefore, the factorization of given cubic polynomial 6x³ - 11x² - 12x + 5 is: 6x³ - 11x² - 12x + 5 = (x + 1)(x - $\frac{5}{2}$ )(x - $\frac{1}{3}$ )

Learn more about the polynomial here:

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2 years ago

Points (−6, 3) and (−6, −3) on the coordinate grid below show the positions of two dogs at a park: A coordinate plane is shown.

Nostrana [21]

Question:

Points (−6, 3) and (−6, −3) on the coordinate grid below show the positions of two dogs at a park.There is a point at (−6, 3) labeled Dog 1. There is a point at (−6, −3) labeled Dog 2.

Which statement best describes the relationship between the positions of the two players?

A.Dog 1's position is Dog 2's position reflected across the y-axis; only the signs of the y-coordinates of Dog 1 and Dog 2 are different.

B.Dog 1's position is Dog 2's position reflected across the x-axis; only the signs of the y-coordinates of Dog 1 and Dog 2 are different.

C.Dog 1's position is Dog 2's position reflected across the x-axis; only the signs of the x-coordinates of Dog 1 and Dog 2 are different.

D. Dog 1's position is Dog 2's position reflected across the y-axis; only the signs of the x-coordinates of Dog 1 and Dog 2 are different.

Answer:

Option B: Dog 1's position is Dog 2's position reflected across the x-axis; only the signs of the y-coordinates of Dog 1 and Dog 2 are different.

Explanation:

The position of the Dog 1 is (−6, 3).

The position of the Dog 2 is (−6, -3).

From the graph, we can see that the image of Dog 1 is reflected over x-axis. Because to reflect an image over x-axis, the signs of y-coordinate must be changed.

Hence, Option B is correct.

Thus, Dog 1's position is Dog 2's position reflected across the x-axis; only the signs of the y-coordinates of Dog 1 and Dog 2 are different.