Find the Roots/Zeros Using the Rational Roots Test x^3-4x^2-7x+10

Problem

x3−4*x2−7*x+10

Solution

1. Identify the constant term and the leading coefficient. The constant term is 10 and the leading coefficient is 1

2. List all possible factors of the constant term p and the leading coefficient q

p:±1,±2,±5,±10

q:±1

3. Determine the possible rational roots p/q by dividing the factors of p by the factors of q

p/q:±1,±2,±5,±10

4. Test the possible roots using synthetic division or substitution. Testing x=1

1−4*(1)2−7*(1)+10=1−4−7+10=0

Since the result is 0 x=1 is a root and (x−1) is a factor.

5. Divide the polynomial by (x−1) using synthetic division to find the remaining quadratic factor.

(x3−4*x2−7*x+10)÷(x−1)=x2−3*x−10

6. Factor the resulting quadratic expression x2−3*x−10

x2−3*x−10=(x−5)*(x+2)

7. Solve for the remaining roots by setting the factors equal to zero.

x−5=0⇒x=5

x+2=0⇒x=−2

Final Answer

x=1,5,−2

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