Question

Find a quadratic polynomial with zeroes - 2 and -1.

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Answer 1

Answer :

x² + 3x + 2

Note :

• The possible values of the variable for which the polynomial becomes zero are called its zeros .

• A quadratic polynomial can have atmost two zeros .

• The general form of a quadratic polynomial is given as ; ax² + bx + c .

• If α and ß are the zeros of the quadratic polynomial ax² + bx + c , then ;
• Sum of zeros , (α + ß) = -b/a
• Product of zeros , (αß) = c/a

• If α and ß are the zeros of a quadratic polynomial , then that quadratic polynomial is given as : k•[ x² - (α + ß)x + αß ] , k ≠ 0.

Solution :

• Given zeros : (-2) and (-1)
• To find : A quadratic polynomial

Let the given zeros of required quadratic polynomial be α = -2 and ß = -1 .

Now ,

Sum of zeros will be given as ;

α + ß = (-2) + (-1) = -2 - 1 = -3

Also ,

Product of zeros will be given as ;

αß = (-2)•(-1) = 2

Now ,

The general quadratic polynomial will be given as ;

=> k•[ x² - (α + ß)x + αß ] , k ≠ 0

=> k•[ x² - (-3)x + 2 ] , k ≠ 0

=> k•[ x² + 3x + 2 ] , k ≠ 0

If k = 1 , then the quadratic polynomial will be ;

x² + 3x + 2

Answer 2

Given:-

• $\sf α = -2$
• $\sf β = -1$ αβ

To find:-

• A quadratic polynomial having these zeroes

Formula:-

$\underline{\boxed{\sf polynomial = x²-(α+β)x+αβ)}}$

Solution:-

:$\implies$ $\sf α+β = -2+(-1)$

:$\implies$ $\sf α+β = -3$

:$\implies$ $\sf αβ = -2(-1)$

:$\implies$ $\sf αβ = 2$

Now, according to the formula, p(x) :-

:$\implies$ $\sf x²-(-3)x+2$

:$\implies$ $\sf x²+3x+2$

hence, x²+3x+2 is the required polynomial.