One factor of $f(x)=5 x^3+5 x^2-170 x+280$ is $(x+7)$. What are all the roots of the function?
A. $x=-4, x=-2$, or $x=7$
B. $x=-7, x=2$, or $x=4$
C. $x=-7, x=5$, or $x=280$
D. $x=-280, x=-5$, or $x=7$
Answer (1)
Divide the cubic function f ( x ) by the given factor ( x + 7 ) to find the quadratic factor q ( x ) .
Solve the quadratic equation q ( x ) = 0 to find the remaining two roots.
Combine the root from the factor ( x + 7 ) and the roots from q ( x ) to find all roots of f ( x ) .
The roots of the function are x = − 7 , x = 2 , x = 4 .
Explanation
Understanding the Problem We are given the cubic function f ( x ) = 5 x 3 + 5 x 2 − 170 x + 280 and told that ( x + 7 ) is a factor. Our goal is to find all the roots of this function. Since we know one factor, we can divide the cubic function by that factor to find a quadratic factor. Then, we can find the roots of the quadratic factor, which will give us the remaining roots of the cubic function.
Finding the Quadratic Factor We know that ( x + 7 ) is a factor of f ( x ) . This means we can write f ( x ) = ( x + 7 ) q ( x ) , where q ( x ) is a quadratic polynomial. To find q ( x ) , we can perform polynomial division or synthetic division. Let's perform polynomial division:
\require e n c l ose 5 x 2 − 30 x + 40 x + 7 \enclose l o n g d i v 5 x 3 + 5 x 2 − 170 x + 280 − ( 5 x 3 + 35 x 2 ) − 30 x 2 − 170 x − ( − 30 x 2 − 210 x ) 40 x + 280 − ( 40 x + 280 ) 0
So, q ( x ) = 5 x 2 − 30 x + 40 .
Solving the Quadratic Equation Now we need to find the roots of the quadratic q ( x ) = 5 x 2 − 30 x + 40 . We can first simplify by dividing by 5: x 2 − 6 x + 8 = 0 . Now we can factor this quadratic: ( x − 4 ) ( x − 2 ) = 0 . This gives us the roots x = 4 and x = 2 .
Finding All Roots The roots of the cubic function f ( x ) are the roots of ( x + 7 ) and the roots of q ( x ) . The root of ( x + 7 ) is x = − 7 , and the roots of q ( x ) are x = 4 and x = 2 . Therefore, the roots of f ( x ) are x = − 7 , x = 2 , and x = 4 .
Final Answer The roots of the function f ( x ) = 5 x 3 + 5 x 2 − 170 x + 280 are x = − 7 , 2 , 4 .
Examples
Understanding the roots of a polynomial can help engineers design stable structures. For example, if f ( x ) represents the stress on a bridge as a function of temperature x , finding the roots tells us at which temperatures the stress is zero, indicating potential failure points. By analyzing the roots, engineers can implement safety measures to prevent structural failures under varying conditions. This ensures the bridge's stability and longevity.
