Ex - 2.2
Question 1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
Solution
So, the zeroes of x² - 2x - 8 are 4 and -2.
Concept insight: The zero of a polynomial is that value of the variable which when substituted in the polynomial makes its value 0.
When a quadratic polynomial is equated to 0, then the values of the variable obtained are the zeroes of that polynomial. The relationship between the zeroes of a quadratic polynomial with its coefficients is very important. Also, while verifying the above relationships, be careful about the signs of the coefficients.
Question 2. Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
Solution
(i) Let the required polynomial be ax² + bx + c, and let its zeroes
and 
Question 1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
Solution
So, the zeroes of x² - 2x - 8 are 4 and -2.
Concept insight: The zero of a polynomial is that value of the variable which when substituted in the polynomial makes its value 0.
When a quadratic polynomial is equated to 0, then the values of the variable obtained are the zeroes of that polynomial. The relationship between the zeroes of a quadratic polynomial with its coefficients is very important. Also, while verifying the above relationships, be careful about the signs of the coefficients.
Question 2. Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
Solution
(i) Let the required polynomial be ax² + bx + c, and let its zeroes
and
If a = 4k, then b = -k, c = -4k
Therefore, the quadratic polynomial is k(4 x 2 - x - 4), where k is a real number .
(ii) Let the polynomial be ax² + bx + c, and let its zeroes be and
(ii) Let the polynomial be ax² + bx + c, and let its zeroes be
and
(iii) Let the polynomial be ax² + bx + c, and let its zeroes be and
and
(iv) Let the polynomial be ax² + bx + c, and let its zeroes be and
and
Therefore, the quadratic polynomial is k(x² - x + 1),where k is a real number .
(v) Let the polynomial be ax² + bx + c, and its zeroes be and
(v) Let the polynomial be ax² + bx + c, and its zeroes be
and
Therefore, the quadratic polynomial is k(4x² + x + 1),where k is a real number .
(vi) Let the polynomial be ax² + bx + c.
(vi) Let the polynomial be ax² + bx + c.
Therefore, the quadratic polynomial is k(x² - 4x + 1),where k is a real number .
Concept insight: Since the sum and product of zeroes gives 2 relations between three unknowns so we assign a value to the variable a and obtain other values.
Alternatively If the sum and the product of the zeroes of a quadratic polynomial is given then polynomial is given by , where k is a constant. And the simplest polynomial will be the one in which k = 1.
Concept insight: Since the sum and product of zeroes gives 2 relations between three unknowns so we assign a value to the variable a and obtain other values.
Alternatively If the sum and the product of the zeroes of a quadratic polynomial is given then polynomial is given by , where k is a constant. And the simplest polynomial will be the one in which k = 1.
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