find the zeros of g (x)=4x^3+13x^2-37x-10

Answer 1

To find the zeros of the polynomial \( g(x) = 4x^3 + 13x^2 - 37x - 10 \), we can use the Rational Root Theorem to test for possible rational roots. The theorem states that any rational solution of the polynomial equation, in the form \( \frac{p}{q} \), must have \( p \) as a factor of the constant term (-10) and \( q \) as a factor of the leading coefficient (4).

Step 1: Possible Rational Roots

• Factors of the constant term (-10): \( \pm 1, \pm 2, \pm 5, \pm 10 \)
• Factors of the leading coefficient (4): \( \pm 1, \pm 2, \pm 4 \)

Possible rational roots can therefore be: \[ \pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm 2, \pm 5, \pm 10 \]

Step 2: Test Possible Rational Roots

Let's test these possible roots by substituting them into \( g(x) \).

1. Testing \( x = 1 \): \[ g(1) = 4(1)^3 + 13(1)^2 - 37(1) - 10 = 4 + 13 - 37 - 10 = -30 \quad (\text{not a root}) \]

2. Testing \( x = -1 \): \[ g(-1) = 4(-1)^3 + 13(-1)^2 - 37(-1) - 10 = -4 + 13 + 37 - 10 = 36 \quad (\text{not a root}) \]

3. Testing \( x = 2 \): \[ g(2) = 4(2)^3 + 13(2)^2 - 37(2) - 10 = 4(8) + 13(4) - 74 - 10 = 32 + 52 - 74 - 10 = 0 \quad (\text{is a root}) \]

So, \( x = 2 \) is one root.

Step 3: Factor Out \( (x - 2) \)

Now we can use synthetic division to factor \( g(x) \) by \( x - 2 \).

\[ \begin{array}{r|rrrr} 2 & 4 & 13 & -37 & -10 \ & & 8 & 42 & 10 \ \hline & 4 & 21 & 5 & 0 \ \end{array} \]

The quotient is \( 4x^2 + 21x + 5 \).

Step 4: Solve \( 4x^2 + 21x + 5 = 0 \)

Now we can apply the quadratic formula: \[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \] Where \( a = 4 \), \( b = 21 \), and \( c = 5 \).

Calculate the discriminant: \[ D = b^2 - 4ac = 21^2 - 4(4)(5) = 441 - 80 = 361 \]

Now apply the quadratic formula: \[ x = \frac{{-21 \pm \sqrt{361}}}{2(4)} = \frac{{-21 \pm 19}}{8} \]

Calculating the two possible roots:

1. \( x = \frac{{-21 + 19}}{8} = \frac{-2}{8} = -\frac{1}{4} \)
2. \( x = \frac{{-21 - 19}}{8} = \frac{-40}{8} = -5 \)

Summary of Roots

The zeros of \( g(x) = 4x^3 + 13x^2 - 37x - 10 \) are: \[ x = 2, \quad x = -\frac{1}{4}, \quad x = -5 \]