question_answer 13)

                (a) Simplify \[3x(4x-5)\,+3\] and find its values for                 (i) \[x=3,\]                 (ii) \[x=\frac{1}{2}\]. (b) Simplify: \[a({{a}^{2}}+a+1)\,+5\] and find its value for (i) \[a=0,\] (ii) a = 1 and ( (iii) a = - 1.

Answer:

                (a) \[3x(4x-5)+3\]                 \[=(3x)\,(4x)-(3x)(5)+3\]                 \[=(3\times 4)\,\times (x\times x)\,-15x+3\]                 (i) When \[x=3,\,12{{x}^{2}}-15x+3\]                 \[=12{{(3)}^{2}}-15(3)+3\] \[=108-45+3\] = 66 (ii) When \[x=\frac{1}{2},\,12{{x}^{2}}-15x+3\] \[=12{{\left( \frac{1}{2} \right)}^{2}}-15\left( \frac{1}{2} \right)+3\] \[=3-\frac{15}{2}+3\] \[=-\frac{3}{2}\] (b) \[a({{a}^{2}}+a+1)+5\]                 \[=a\times {{a}^{2}}+a\times a+a\times 1+5\] \[={{a}^{3}}+{{a}^{2}}+a+5\]                 (ii) When a = 1 \[{{a}^{3}}+{{a}^{2}}+a+5={{(1)}^{3}}+{{(1)}^{2}}+(1)+5\] \[=1+1+1+5\] = 8                 (iii) When a = - 1 \[{{a}^{3}}+{{a}^{2}}+a+5\] \[={{(-1)}^{3}}+{{(-1)}^{2}}+(-1)+5\] \[=-1+1-1+5\] = 8                 (iii) When a = - 1                 \[{{a}^{3}}+{{a}^{2}}+a+5\] \[={{(-1)}^{3}}+{{(-1)}^{2}}+(-1)+5\]  \[=-1+1-1+5\] = 4.