Question

Find the zeros of the quadratic polynomial \[f\left( x \right) = {x^2} - 3x - 28\] and verify the relationships between the zeros and the coefficients.
A) \[x = 7,{\text{ }} - 4\]
B) \[{\text{sum of roots}} = - 3\]
C) \[{\text{product of roots}} = - 28\]
D) \[x = - 7,4\]

Answer 1

Hint: First we will take the given quadratic polynomial equal to 0 and then we will factorize the obtained equation to find the zeros of the given polynomial. Then we will find the sum and product of the obtained roots to find the correct options.

Complete step by step solution:
We are given that the quadratic polynomial \[f\left( x \right) = {x^2} - 3x - 28\].

First, we will take the given quadratic polynomial equal to 0.

\[ \Rightarrow {x^2} - 3x - 28 = 0\]

Factorizing the above equation, we get

\[
   \Rightarrow {x^2} - 7x + 4x - 28 = 0 \\
   \Rightarrow x\left( {x - 7} \right) + 4\left( {x - 7} \right) = 0 \\
   \Rightarrow \left( {x + 4} \right)\left( {x - 7} \right) = 0 \\
 \]
\[ \Rightarrow x + 4 = 0\] or \[ \Rightarrow x - 7 = 0\]
\[ \Rightarrow x = - 4\] or \[x = 7\]

Thus, the zeros of the given quadratic polynomial \[f\left( x \right)\] are \[ - 4\], 7.
Hence, option A is correct and option D is incorrect.

Now, we will find the sum of the above roots of the given quadratic polynomial.

\[ \Rightarrow - 4 + 7 = 3\]
Hence, option B is correct.

Multiplying the above zeros of the given quadratic polynomial, we get

\[ \Rightarrow - 4 \times 7 = - 28\]
Hence, option C is correct.

Therefore, the options A, B and C are correct.

Note: In solving these types of questions, you should be familiar with the formula of factorization of quadratic equations. It doesn’t matter in which order you use the two numbers. One should know that a quadratic polynomial is a polynomial with one or more variables In which the highest-degree term is of the second degree. If the quadratic equation is equal to zero, then the solutions to the equation are called the root of the univariate function. Students should take care of the fact that this problem has multiple correct options to find real answers.