Educational

Find x³ − 1/x³ Question: \[ x – \frac{1}{x} = 3 + 2\sqrt{2} \] Find: \[ x^3 – \frac{1}{x^3} \] Solution: Using identity: \[ \left(a-b\right)^3 = a^3-b^3-3ab(a-b) \] \[ \left(x-\frac{1}{x}\right)^3 = x^3-\frac{1}{x^3} – 3\left(x\right)\left(\frac{1}{x}\right) \left(x-\frac{1}{x}\right) \] Since, \[ x\cdot \frac{1}{x}=1 \] \[ \left(x-\frac{1}{x}\right)^3 = x^3-\frac{1}{x^3} – 3\left(x-\frac{1}{x}\right) \] \[ x^3-\frac{1}{x^3} = \left(x-\frac{1}{x}\right)^3 + 3\left(x-\frac{1}{x}\right) \]

Find the Product (x³ + 1)(x⁶ − x³ + 1) Find the Product: \[ (x^3 + 1)(x^6 – x^3 + 1) \] Solution: Using identity: \[ (a+b)(a^2-ab+b^2)=a^3+b^3 \] Here, \[ a=x^3,\qquad b=1 \] \[ (x^3 + 1)(x^6 – x^3 + 1) \] \[ = (x^3 + 1)\left[(x^3)^2 – (x^3)(1) + 1^2\right] \] \[ = (x^3)^3

Find the Product (x² − 1)(x⁴ + x² + 1) Find the Product: \[ (x^2 – 1)(x^4 + x^2 + 1) \] Solution: Using identity: \[ (a-b)(a^2+ab+b^2)=a^3-b^3 \] Here, \[ a=x^2,\qquad b=1 \] \[ (x^2 – 1)(x^4 + x^2 + 1) \] \[ = (x^2 – 1)\left[(x^2)^2 + (x^2)(1) + 1^2\right] \] \[ = (x^2)^3

Find the Product (1 + x)(1 − x + x²) Find the Product: \[ (1 + x)(1 – x + x^2) \] Solution: Using identity: \[ (a+b)(a^2-ab+b^2)=a^3+b^3 \] Here, \[ a=1,\qquad b=x \] \[ (1 + x)(1 – x + x^2) \] \[ = (1 + x)(1^2 – 1\cdot x + x^2) \] \[ =

Find the Product (1 − x)(1 + x + x²) Find the Product: \[ (1 – x)(1 + x + x^2) \] Solution: Using identity: \[ (a-b)(a^2+ab+b^2)=a^3-b^3 \] Here, \[ a=1,\qquad b=x \] \[ (1 – x)(1 + x + x^2) \] \[ = (1 – x)(1^2 + 1\cdot x + x^2) \] \[ =

Find the Product (3/x − 2x²)(9/x² + 4x⁴ + 6x) Find the Product: \[ \left(\frac{3}{x} – 2x^2\right) \left(\frac{9}{x^2} + 4x^4 + 6x\right) \] Solution: Rearranging the terms: \[ \left(\frac{3}{x} – 2x^2\right) \left(\frac{9}{x^2} + \frac{6x^3}{x^2} + 4x^4\right) \] \[ = \left(\frac{3}{x} – 2x^2\right) \left(\frac{9}{x^2} + 6x + 4x^4\right) \] Using identity: \[ (a-b)(a^2+ab+b^2)=a^3-b^3 \] Here, \[

Find the Product (2/x + 3x)(4/x² + 9x² − 6) Find the Product: \[ \left(\frac{2}{x} + 3x\right) \left(\frac{4}{x^2} + 9x^2 – 6\right) \] Solution: Rearranging the terms: \[ \left(\frac{2}{x} + 3x\right) \left(\frac{4}{x^2} – \frac{6}{1} + 9x^2\right) \] Using identity: \[ (a+b)(a^2-ab+b^2)=a^3+b^3 \] Here, \[ a=\frac{2}{x},\qquad b=3x \] \[ \left(\frac{2}{x} + 3x\right) \left[\left(\frac{2}{x}\right)^2 – \left(\frac{2}{x}\right)(3x) +

Find the Product (3 + 5/x)(9 − 15/x + 25/x²) Find the Product: \[ \left(3 + \frac{5}{x}\right) \left(9 – \frac{15}{x} + \frac{25}{x^2}\right) \] Solution: Using identity: \[ (a+b)(a^2-ab+b^2)=a^3+b^3 \] Here, \[ a=3,\qquad b=\frac{5}{x} \] \[ \left(3 + \frac{5}{x}\right) \left(9 – \frac{15}{x} + \frac{25}{x^2}\right) \] \[ = (3)^3 + \left(\frac{5}{x}\right)^3 \] \[ = 27 + \frac{125}{x^3}

Find the Product (3/x − 5/y)(9/x² + 25/y² + 15/xy) Find the Product: \[ \left(\frac{3}{x} – \frac{5}{y}\right) \left(\frac{9}{x^2} + \frac{25}{y^2} + \frac{15}{xy}\right) \] Solution: Using identity: \[ (a-b)(a^2+ab+b^2)=a^3-b^3 \] Here, \[ a=\frac{3}{x},\qquad b=\frac{5}{y} \] \[ \left(\frac{3}{x} – \frac{5}{y}\right) \left(\frac{9}{x^2} + \frac{15}{xy} + \frac{25}{y^2}\right) \] \[ = \left(\frac{3}{x}\right)^3 – \left(\frac{5}{y}\right)^3 \] \[ = \frac{27}{x^3} – \frac{125}{y^3}

Find the Product (x/2 + 2y)(x²/4 − xy + 4y²) Find the Product: \[ \left(\frac{x}{2} + 2y\right)\left(\frac{x^2}{4} – xy + 4y^2\right) \] Solution: Using identity: \[ (a+b)(a^2-ab+b^2)=a^3+b^3 \] Here, \[ a=\frac{x}{2},\qquad b=2y \] \[ \left(\frac{x}{2} + 2y\right)\left(\frac{x^2}{4} – xy + 4y^2\right) \] \[ = \left(\frac{x}{2} + 2y\right) \left[\left(\frac{x}{2}\right)^2 – \left(\frac{x}{2}\right)(2y) + (2y)^2\right] \] \[ =