Which values of [tex]$x$[/tex] would make a polynomial equal to zero if the factors of the polynomial were [tex]$(x+6)$[/tex] and [tex]$(x+9)$[/tex]?
A. 6 and -9
B. -6 and -9
C. 6 and 9
D. -6 and 9
Answer (2)
Set each factor to zero: x + 6 = 0 and x + 9 = 0 .
Solve for x in each equation: x = − 6 and x = − 9 .
The values of x that make the polynomial zero are − 6 and − 9 .
The final answer is − 6 , − 9 .
Explanation
Understanding the Problem We are given a polynomial with factors ( x + 6 ) and ( x + 9 ) . We want to find the values of x that make the polynomial equal to zero. This means we need to solve the equation ( x + 6 ) ( x + 9 ) = 0 .
Setting Factors to Zero To solve the equation ( x + 6 ) ( x + 9 ) = 0 , we set each factor equal to zero and solve for x .
Solving for x in the First Factor First, we solve x + 6 = 0 . Subtracting 6 from both sides, we get x = − 6 .
Solving for x in the Second Factor Next, we solve x + 9 = 0 . Subtracting 9 from both sides, we get x = − 9 .
Finding the Values of x Therefore, the values of x that make the polynomial equal to zero are x = − 6 and x = − 9 .
Final Answer The values of x that make the polynomial equal to zero are − 6 and − 9 .
Examples
Understanding how to find the roots of a polynomial by factoring is crucial in many real-world applications. For instance, engineers use this concept to design stable structures. Imagine designing a bridge; the polynomial's roots can represent critical stress points. By ensuring these points are within safe limits, the bridge's structural integrity is maintained. Similarly, in physics, finding the zeros of a function can help determine equilibrium points in a system, ensuring stability and predictability.
The values of x that make the polynomial equal to zero are − 6 and − 9 , thus the correct answer is B. -6 and -9.
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