1. Find the square root of the following polynomials by factorization method
(i) \( x^2 – 8x + 16 \)
\[x^2 – 8x + 16 = (x – 4)^2\]
\[\sqrt{x^2 – 8x + 16} = x – 4\]
(ii) \( 9x^2 + 12x + 4 \)
\[9x^2 + 12x + 4 = (3x + 2)^2\]
\[\sqrt{9x^2 + 12x + 4} = 3x + 2\]
(iii) \( 36a^2 + 84a + 49 \)
\[36a^2 + 84a + 49 = (6a + 7)^2\]
\[\sqrt{36a^2 + 84a + 49} = 6a + 7\]
(iv) \( 64y^2 – 32y + 4 \)
\[64y^2 – 32y + 4 = (8y – 2)^2\]
\[\sqrt{64y^2 – 32y + 4} = 8y – 2\]
(v) \( 200t^2 – 120t + 18 \)
\[200t^2 – 120t + 18 = 2(100t^2 – 60t + 9)\]
\[100t^2 – 60t + 9 = (10t – 3)^2\]
\[\sqrt{200t^2 – 120t + 18} = \sqrt{2}(10t – 3)\]
(vi) \( 40x^2 + 120x + 90 \)
\[40x^2 + 120x + 90 = 10(4x^2 + 12x + 9)\]
\[4x^2 + 12x + 9 = (2x + 3)^2\]
\[\sqrt{40x^2 + 120x + 90} = \sqrt{10}(2x + 3)\]
2. Find the square root of the following polynomials by division method
(i) \( 4x^4 – 28x^3 + 37x^2 + 42x + 9 \)
\[\sqrt{4x^4 – 28x^3 + 37x^2 + 42x + 9} = (2x^2 – 7x + 3)\]
(ii) \( 121x^4 – 198x^3 – 183x^2 + 216x + 144 \)
\[\sqrt{121x^4 – 198x^3 – 183x^2 + 216x + 144} = (11x^2 – 9x – 12)\]
(iii) \( x^4 – 10x^3y + 27x^2y^2 – 10xy^3 + y^4 \)
\[x^4 – 10x^3y + 27x^2y^2 – 10xy^3 + y^4 = (x^2 – xy + y^2)^2\]
\[\sqrt{x^4 – 10x^3y + 27x^2y^2 – 10xy^3 + y^4} = x^2 – xy + y^2\]
(iv) \( 4x^4 – 12x^3 + 37x^2 – 42x + 49 \)
\[\sqrt{4x^4 – 12x^3 + 37x^2 – 42x + 49} = (2x^2 – 3x + 7)\]
3. An investor’s return \( R(x) = -x^2 + 6x – 8 \). Factorize the expression and find the investment levels that result in zero return.
\[R(x) = -x^2 + 6x – 8\]
\[R(x) = -(x^2 – 6x + 8)\]
\[x^2 – 6x + 8 = (x – 4)(x – 2)\]
\[R(x) = -(x – 4)(x – 2)\]
\[x = 4, x = 2\]
4. A company’s profit \( P(x) = x^3 – 15x^2 + 75x – 125 \). Find the break-even point(s), where the profit is zero.
\[P(x) = x^3 – 15x^2 + 75x – 125\]
\[P(x) = (x – 5)(x^2 – 10x + 25)\]
\[x^2 – 10x + 25 = (x – 5)^2\]
\[P(x) = (x – 5)^3\]
\[x = 5\]
5. The potential energy \( V(x) = 2x^3 – 6x^2 + 4x \). Determine where the potential energy is zero.
\[V(x) = 2x(x^2 – 3x + 2)\]
\[x^2 – 3x + 2 = (x – 1)(x – 2)\]
\[V(x) = 2x(x – 1)(x – 2)\]
\[x = 0, x = 1, x = 2\]
6. In structural engineering, the deflection \( Y(x) = 2x^2 – 8x + 6 \). Find the points of zero deflection.
\[Y(x) = 2(x^2 – 4x + 3)\]
\[x^2 – 4x + 3 = (x – 3)(x – 1)\]
\[Y(x) = 2(x – 3)(x – 1)\]
\[x = 3, x = 1\]
