Question

Write the second degree polynomial $x^{2} + 5 x - 84$ as the product of two first degree polynomials.

Accepted Answer

The given second degree polynomial is $x^{2} + 5 x - 84$ .

We need to write the given second degree polynomial as the product of two first degree polynomials.

If $x - a$ is a factor of $p (x)$ , then $p (a) = 0$ . In other words, $a$ must be a solution of the equation $x^{2} + 5 x - 84 = 0$

By, comparing $x^{2} + 5 x - 84 = 0$ with standard form of quadratic equation $a x^{2} + b x + c = 0$ , we get $a = 1$ , $b = 5$ and $c = - 84$

We know that, formula to solve the quadratic equation is $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

Thus, $x = \frac{- (5) \pm \sqrt{{(5)}^{2} - 4 \times 1 \times (- 84)}}{2 \times 1}$

$\Rightarrow x = \frac{- 5 \pm \sqrt{25 + 336}}{2} = \frac{- 5 \pm \sqrt{361}}{2}$

$\Rightarrow x = \frac{- 5 \pm 19}{2}$

$\Rightarrow x = \frac{- 5 + 19}{2}, \frac{- 5 - 19}{2}$

$\Rightarrow x = \frac{14}{2} = 7, \frac{- 24}{2} = - 12$

This means $p (7) = 0$ and $p (- 12) = 0$

From this, it follows that $x - 7$ and $x + 12$ are factors of $p (x)$ .

Therefore, $x^{2} + 5 x - 84 = (x - 7) (x + 12)$ .

Kerala Board Solutions for Chapter: Polynomials, Exercise 3: Exercise 3

Kerala Board Mathematics Solutions for Exercise - Kerala Board Solutions for Chapter: Polynomials, Exercise 3: Exercise 3

Questions from Kerala Board Solutions for Chapter: Polynomials, Exercise 3: Exercise 3 with Hints & Solutions

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