Question

Solve each polynomial equation.

15. 15x^3-119x^2-10x+16=0
16. x^3-14x^2+47x-18=0
17. 5x^3-27x^2-17x-6=0

Answer 1

15. 8, - 0.4 and 0.333 to nearest thousandth.

Explanation:

15. as the last value is 16 we might try x = 4 and x = 8 as roots

f(4) = -968 - Not a root . f(8) = 0 so x = 8 is one root and x - 8 is a factor

If we divide the function by x - 8 we get 15x^2 + x - 2

which factors to (5x + 2)(3x - 1)

(5x + 2)(3x - 1) = 0 giving x = -0.4 and x = 1/3.

So the roots are 8, -0.4 and 1/3

Answer 2

Answers:

15) x = 8,
`(1)/(3), - (2)/(5)`

16) x = 9,
`\frac{5+√(17)} {2}`,
`\frac{5 -√(17)} {2}`

17) x = 6,
`-\frac{3+i√(11)} {10}`,
`-\frac{3-i√(11)} {10}`

Explanation:

15x³ - 119x² - 10x + 16 = 0

`(p)/(q): (16)/(15):+/- (1*2*4*8*16)/(1*3*5*15)`

So, the possible rational roots are: +/-
`1, 2, 4, 8, 16,(1)/(3),(2)/(3),(4)/(3),(8)/(3),(16)/(3),(1)/(5),(2)/(5),(4)/(5),(8)/(5),(16)/(5),(1)/(15),(2)/(15),(4)/(15),(8)/(15),(16)/(15)`

Use synthetic division with each one until you find a remainder of zero. I am not going to go through each one because it is too time consuming, however, the first one that works is x = 8

(x - 8)(15x² + x - 2)

Next, factor 15x² + x - 2 using any method

(x - 8)(3x - 1)(5x + 2)

Now, solve for x.

x = 8, x =
`(1)/(3)`, x =
`-(2)/(5)`

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For #16 & 17, follow the same process as above.