Question

16. If 2 and -2 are two zeros of the polynomial 2x^4 -5x^3 -11x^2+ 20x + 12,find all the zeros of the given polynomial.
[CBSE 2019)​

Answer 1

Given:-

• The two zeros of p(x)= 2, -2
• P(x)= 2x⁴- 5x³ - 11x² + 20x +12

To Find:-

• The others zeros of P(x)=?

Solution:-

• To calculate the other zeros of polynomial at first we have to find the factors. with the help of given zeros we have to find factors . As given in the question zeros are 2, - 2. Here (x+2)(x-2) as in the form of x²-4 is the factors of the given polynomial. Now dividing P(x) by x²-4 to find quardric polynomial.

Let,s dividing p(x) by x² - 4 :-

x² - 4 ) 2x⁴ -5x³ - 11x² + 20x + 12 ( 2x² -5x -3

2x⁴ - 8x²

____-_______+_______________

-5x³ - 3x² + 20x + 12

-5x³. +20x

_________+_______-____________

-3x² + 12

-3x² + 12

_____________+____-______________

0

By dividing p(x) by x² - 4 we get here:-

==> P(x) = (x²-4) (2x² - 5x - 3)

==> (x²- 4) (2x² -6x + x - 3)

==> (x²- 4) { 2x (x - 3) + 1 ( x - 3) }

==> ( x² - 4) (2x + 1)(x - 3)

==> (2x + 1) (x + 3) = 0

==> 2x + 1 = 0 , => 2x = -1 , => x = -1 / 2

==> x - 3 = 0 , => x = 3

Hence,

• The other zeros of p(x)= -1 / 2 , 3

Answer 2

$\bold{\purple{\underline{\underline{Given \: that,}}}}$

• 2 and -2 are two zeros of polynomial
• Polynomial = 2x^4 -5x^3 -11x^2+ 20x + 12

$\bold{\purple{\underline{\underline{To \: find:-}}}}$

• All the zeroes of polynomial

$\bold{\purple{\underline{\underline{As \: we \: know:-}}}}$

• (a-b) (a+b) = a²-b²
• g (x) = 0

$\bold{\purple{\underline{\underline{Concept:-}}}}$

• Polynomials

Understanding the concept...

Polynomials are a group of expressions which consists of some variables and some coefficients. Which are used for only a specific purpose that is of multiplication, division, subtraction.

Now let's understand the way to solve:

1. While solving a equation first put in standard form with 0 on one side.

2. Now simplify the equation

3. Find one factor in the given equation.

4. Now divide that factor.

And solve it

$\bold{\purple{\underline{\underline{Step \: by \: step \: explaination:-}}}}$

Step 1.

Taking the factors that is,

p (x) = (x-2) (x+2)

Step 2.

Dividing the given polynomial p(x) by x² -4.

✔️ Refer the attachment

Step 3.

Evaluating values in g(x)

→ 2x² - 5x - 3 = 0

Splitting middle term.

→ 2x² - 6x + x - 3

Grouping the terms.

→ 2x (x-3) + (x-3)= 0

→ (x-3) (2x + 1) = 0

→ x = 3, -1/2