Question 2 of 2intermediate🔧 ApplyShort Answer2 marks

Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.

(i)

14,−1\frac{1}{4}, -1

Answer

4x2−x−44x^2 - x - 4

(ii)

2,13\sqrt{2}, \frac{1}{3}

Answer

3x2−32x+13x^2 - 3\sqrt{2}x + 1

(iii)

0,50, \sqrt{5}

Answer

x2+5x^2 + \sqrt{5}

(iv)

1,11, 1

Answer

x2−x+1x^2 - x + 1

(v)

−14,14-\frac{1}{4}, \frac{1}{4}

Answer

4x2+x+14x^2 + x + 1

(vi)

4,14, 1

Answer

x2−4x+1x^2 - 4x + 1

Explanation

This question tests the relationship between zeroes and coefficients of a quadratic polynomial. From the textbook context, for a quadratic polynomial ax2+bx+cax^2 + bx + c with zeroes α\alpha and β\beta, the sum of zeroes equals −b/a-b/a and the product equals c/ac/a. Taking a=1a = 1, we get b=−(sum)b = -(\text{sum}) and c=productc = \text{product}, giving the polynomial x2−(sum)x+(product)x^2 - (\text{sum})x + (\text{product}). Multiples of these polynomials are also valid answers.