Which values of $x$ would make a polynomial equal to zero if the factors of the polynomial were $(x-3)$ and $(x-7)$?
A. 3 and 7
B. 3 and -7
C. -3 and 7
D. -3 and -7
Answer (1)
Set each factor to zero: x − 3 = 0 and x − 7 = 0 .
Solve for x in each equation: x = 3 and x = 7 .
The values of x that make the polynomial equal to zero are 3 and 7.
The answer is 3 and 7
Explanation
Understanding the Problem We are given the factors of a polynomial, ( x − 3 ) and ( x − 7 ) , and we want to find the values of x that make the polynomial equal to zero. This means we want to find the roots of the polynomial.
Setting Factors to Zero To find the values of x that make the polynomial equal to zero, we set each factor equal to zero and solve for x .
Solving for x (First Factor) First, we set the factor ( x − 3 ) equal to zero: x − 3 = 0 Adding 3 to both sides, we get: x = 3
Solving for x (Second Factor) Next, we set the factor ( x − 7 ) equal to zero: x − 7 = 0 Adding 7 to both sides, we get: x = 7
Final Answer Therefore, the values of x that make the polynomial equal to zero are 3 and 7.
Examples
Understanding how to find the roots of a polynomial by factoring is crucial in many areas, such as physics and engineering. For example, when analyzing the trajectory of a projectile, the roots of a quadratic equation representing the height of the projectile can tell us when the projectile hits the ground. If the height equation is h ( t ) = ( t − 2 ) ( t − 8 ) , the roots t = 2 and t = 8 represent the times when the projectile is at ground level. This helps engineers design systems and predict outcomes in real-world scenarios.
