approximate to the nearest tenth the real zeros of (fx) = -5X^3 = 9X^2 + 12X + 2

Question

approximate to the nearest tenth the real zeros of (fx) = -5X³ = 9X² + 12X + 2

Answer 1

-5X³ = 9X² + 12X + 2

Taking the derivative set to 0 yields the critical numbers:

-15x² = 18x + 12

zeros at (-.5,0) and (1.7,0) using the quadratic formulas

Answer 2

Hi Cheryl,

I think you must have a typo in the problem, and that second = sign should be a + sign? I'll solve assuming that.

Step 1) Find factors of the constant term (2) and the leading coefficient (-5).

Factors of the constant term are 1 and 2.

Factors of the leading coefficient are 1 and -5.

Step 2) We want to find one of the roots, so we can divide it out using synthetic division. One of the roots will be one of the following:

± (a factor of the constant term) / (a factor of the leading coefficient)

So we have eight possibilities:

±1, ±1/5, ±2, ±2/5

Step 3) Try these possibilities one by one until you find the one that works. It is always easiest to start with 1 or -1.

For example, if 1 is a root, then (x-1) is a term we can factor out. Let's try (x-1) first.

1 | -5 9 12 2

| ↓ -5 4 16

-5 4 16 18

The remainder from synthetic division is 18. Since it is not zero, that means (x-1) is NOT a factor.

So we need to try again.

If -1 is a root, then (x+1) is a term we can factor out. Let's try (x+1) next.

-1 | -5 9 12 2

| ↓ 5 -14 2

-5 14 -2 4

The remainder from synthetic division is 4. Since it is not zero, that means (x+1) is NOT a factor.

So we need to try again.

If -1/5 is a root, then (x+1/5) is a term we can factor out. Let's try (x+1/5) next.

-1/5 | -5 9 12 2

| ↓ 1 -2 -2

-5 10 10 0

The remainder from synthetic division is 0. Since it IS zero, that means (x+1/5) IS a factor!

From synthetic division, we know:

(-5x³+9x²+12x+2) ÷ (x+1/5) = (-5x²+10x+10)

Step 4) Look at newly factored function and try to factor further on the trinomial.

f(x) = (x+1/5)(-5x²+10x+10)

The trinomial doesn't factor easily so we need to use quadratic formula.

(-b ± (b²-4ac)^1/2 )/(2a) where a=-5, b=10, c=10

Roots are = (-10 ± (10²-4*-5*10)^1/2 )/(2*-5)

Using a calculator,

x= -0.7320508076

x= +2.732050808

So, the entire function can be re-written as the product of its factors:

f(x)=(x+1/5)(x+0.7320508076)(x-2.732050808)

The three roots, rounded to tenths place, are:

x= -0.2

x= -0.7

x= 2.7

Hope this helps! :)