Question

...........................answer asap​

Answer 1

${6x}^{2} - 5x - 6x$

${6x}^{2} - 9x + 4x - 6x$

$3x(2x - 3) + 2x(2x - 3)$

$(3x + 2x)(2x - 3)$

second question

${x}^{2} - 5x + 6$

${x}^{2} - 3x - 2x + 6$

$x(x - 3) - 2(x - 3)$

$(x - 2)(x - 3)$

Answer 2

$\color{red}\large\underline{\underline{Question:}}$

$\text{Find the sum and product of the zeroes}$

$\text{of the polynomials}$

• $6x^{2} - 5x - 6$ [Equation...1]
• $x^{2} - 5x + 6$ [Equation...2]

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$\color{blue}\large\underline{\underline{To\:Find:}}$

$\underline{Sum\:and\:product\:of\:the\:zeroes\:of\:the\:polynomials}$

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$\color{purple}\large\underline{\underline{Concept:}}$

By middle-splitting theorem..

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$\color{green}\large\underline{\underline{Solution:}}$

$\underline{\underline{(For\:Equation\:1):}}$

$6x^{2} - 5x - 6$

$\Rightarrow 6x^{2} - (9 - 4)x - 6$

$\Rightarrow 6x^{2} - 9x + 4x - 6$

$\Rightarrow 3x(2x - 3) + 2(2x - 3)$

$\Rightarrow (2x - 3)(2x + 3)$

________|.....................................................|________

$\text{Zeroes of this polynomial is}$

• $2x - 3 = 0$

$2x = 3$

$\rightarrow x = \dfrac{3}{2}$

• $2x + 3 = 0$

$2x = -3$

$\rightarrow x = -\dfrac{3}{2}$

${\boxed{\therefore x = \dfrac{3}{2} ; -\dfrac{3}{2}}}$

$\underline{Sum\:of\:the\:polynomials}$

$\bigg[\dfrac{3}{2} + \bigg(-\dfrac{3}{2}\bigg)\bigg]$

LCM of denominators is 2..

$\bigg(\dfrac{3 - 3}{2}\bigg)$

$\bigg(\dfrac{0}{2}\bigg)$

$\Rightarrow 0$

$\underline{Product\:of\:the\:polynomials}$

$\bigg[\dfrac{3}{2} × \bigg(-\dfrac{3}{2}\bigg)\bigg]$

$-\bigg(\dfrac{9}{4}\bigg)$

${\boxed{\therefore sum = 0 ; product = -\bigg(\dfrac{9}{4}\bigg)}}$

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$\underline{\underline{(For\:Equation\:2):}}$

$x^{2} - 5x + 6$

$\Rightarrow x^{2} - (3 + 2)x + 6$

$\Rightarrow x^{2} - 3x - 2x + 6$

$\Rightarrow x(x - 3) - 2(x - 3)$

$\Rightarrow (x - 3)(x - 2)$

________|.....................................................|________

$\text{Zeroes of this polynomial is}$

• $x - 3 = 0$

$\Rightarrow x = 3$

• $x - 2 = 0$

$\Rightarrow x = 2$

${\boxed{\therefore x = 3 ; 2}}$

$\underline{Sum\:of\:the\:polynomials}$

$\Rightarrow 3 + 2 = 5$

$\underline{Product\:of\:the\:polynomials}$

$\Rightarrow 3 × 2 = 6$

${\boxed{\therefore sum = 5 ; product = 6}}$

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