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Form a quadratic poly nomial One of whose zero is 2+sqrt3 & sum of zeroes is 4 .

Form a quadratic poly nomial One of whose zero is 2+3 2+\sqrt{3} &\& sum of zeroes is 44 .

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Q. Form a quadratic poly nomial One of whose zero is 2+3 2+\sqrt{3} &\& sum of zeroes is 44 .
  1. Identify Zero and Sum: Identify the given zero and the sum of the zeroes.\newlineWe are given one zero of the quadratic polynomial as 2+32+\sqrt{3} and the sum of the zeroes as 44.
  2. Find Other Zero: Find the other zero using the sum of the zeroes.\newlineIf the sum of the zeroes is 44 and one of the zeroes is 2+32+\sqrt{3}, then the other zero, let's call it 'b', can be found using the equation:\newlineSum of zeroes = Zero 11 + Zero 22\newline4=(2+3)+b4 = (2+\sqrt{3}) + b\newlineNow, solve for 'b':\newlineb=4−(2+3)b = 4 - (2+\sqrt{3})\newlineb=2−3b = 2 - \sqrt{3}
  3. Write Quadratic Polynomial: Write the quadratic polynomial using the zeroes.\newlineThe quadratic polynomial can be written in its factored form as:\newlinef(x)=(x−Zero 1)(x−Zero 2)f(x) = (x - \text{Zero 1})(x - \text{Zero 2})\newlineSubstitute the zeroes into the equation:\newlinef(x)=(x−(2+3))(x−(2−3))f(x) = (x - (2+\sqrt{3}))(x - (2-\sqrt{3}))
  4. Expand Factored Form: Expand the factored form to get the standard form of the quadratic polynomial.\newlinef(x)=(x−2−3)(x−2+3)f(x) = (x - 2 - \sqrt{3})(x - 2 + \sqrt{3})\newlineNow, use the difference of squares formula, which states that (a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2:\newlinef(x)=(x−2)2−(3)2f(x) = (x - 2)^2 - (\sqrt{3})^2\newlinef(x)=(x2−4x+4)−3f(x) = (x^2 - 4x + 4) - 3\newlinef(x)=x2−4x+4−3f(x) = x^2 - 4x + 4 - 3\newlinef(x)=x2−4x+1f(x) = x^2 - 4x + 1

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