What is the completely factored form of f(x) = 6x3 – 13x2 – 4x + 15?

f(x) = (x + 1)(2x - 3)(3x - 5)

Step-by-step explanation:

The given function is f(x) = 6x³- 13x²- 4x + 1

If we substitute x = -1 then we get f(-1) = 0. Therefore (x + 1) is a zero factor.

Now we will use synthetic division to get the polynomial factorized.

(-1) |   6    -13   -4    15  

        ___-6__19_-15_            

        6    -19   15    0

Now we get the polynomial as (x + 1)(6x²- 19x + 15)

Now we factorize (6x² - 19x + 15)

6x² - 19x + 15 = 6x² - 10x - 9x + 15

= 2x(3x - 5) - 3(3x - 5)

= (2x - 3)(3x - 5)

Finally we get the expression in the factored form as (x + 1)(2x - 3)(3x - 5).

  9 nickels

Step-by-step explanation:

Changing a quarter to a nickel decreases the value of the sum by $0.20.

When all quarters are changed to nickels and vice versa, the total value will change by the number of excess quarters. That number is ...

  ($5.95 -3.35)/(0.20) = 2.60/0.20 = 13

The remaining change is made up of equal numbers of nickels and quarters. That amount of change is ...

  $5.95 -13·0.25 = 2.70

A quarter and nickel together total $0.30, so there must be $2.70/$0.30 = 9 such pairs of coins.

There are 9 nickels and 22 quarters.

If you don't want to reason through the problem, you can write equations. Let n and q represent the numbers of nickels and quarters, respectively. Then you have ...

  0.05n + 0.25q = 5.95

  0.25n + 0.05q = 3.35

Multiplying the second equation by 5 and subtracting the first gives ...

  5(0.25n +0.05q) -(0.05n +0.25q) = 5(3.35) -(5.95)

  1.20n = 10.80

  10.80/1.20 = n = 9

The number of nickels is 9.