EXERCISE – 3.3

1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

(i) x ² – 2x – 8      (ii) 4s ² – 4s + 1      (iii) 6x ² – 3 – 7x      (iv) 4u ² + 8u     (v) t² – 15          (vi) 3x ²2 – x – 4

2. Find the quadratic polynomial in each case, with the given numbers as the sum and product of its zeroes respectively.

(i) -1/ 4 , – 1         (ii) √2,1/ 3             (iii) 0, √5    (iv) 1, 1            (v) – 1/ 4 , 1 4         (vi) 4, 1

3. Find the quadratic polynomial, for the zeroes a, b given in each case.

(i) 2, –1    (ii) √3 , √– 3    (iii) 1/4,-1   (iv) 1/ 2 , 3/ 2

4. Verify that 1, –1 and +3 are the zeroes of the cubic polynomial x ³ – 3x ² – x + 3 and check the relationship between zeroes and the coefficients.

3.7 DIVISION ALGORITHM FOR POLYNOMIALS

You know that a cubic polynomial has at most three zeroes. However, if you are given only one zero, can you find the other two? For example, let us consider the cubic polynomial x ³ + 3x ² – x – 3. If one of its zeroes is 1, then you know that this polynomial is divisible by x – 1. Therefore dividing by x – 1 we would get the quotient x ² – 2x – 3.

We get the factors of x ² – 2x – 3 by splitting the middle term. The factors are (x + 1) and (x – 3). This gives us

x ³ – 3x ² – x + 3 = (x – 1) (x ² – 2x – 3)

= (x – 1) (x + 1) (x – 3)

So, the three zeroes of the cubic polynomial are 1, – 1, 3.

Let us discuss the method of dividing one polynomial by another in detail. Before doing the steps formally, consider an example.



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