Question

27. Se ab = 8 e a²b + ab²+ a + b = 90, então qual o valor de a³+ b³?

a) 760.

b) 750.

c) 740.

d) 860.

e) 840.

Answer 1

Fatoração do cubo da diferença de dois termos

${a}^{3}+{b}^{3}\\=(a+b)({a}^{2}-ab+{b}^{2})$

${a}^{2}b+a{b}^{2}+a+b=90\\ab(a+b)+(a+b) =(a+b)(ab+1)$

${a}^{2}b+a{b}^{2}=(a+b)(8+1)=90\\9(a+b)=90\\(a+b)=\frac{90}{9}\\a+b=10$

${(a+b)}^{2}={a}^{2}+2ab+{b}^{2}\\{a}^{2}+{b}^{2}={(a+b)}^{2}-2ab$

${a}^{2}+{b}^{2}={10}^{2}-2.8=100-16=84$

${a}^{3}+{b}^{3}=(a+b)({a}^{2}-ab+{b}^{2})$

${a}^{3}+{b}^{3}=(a+b)({a}^{2}+{b}^{2}-ab)$

${a}^{3}+{b}^{3}=10(84-8)$

${a}^{3}+{b}^{3}=10.76$

${a}^{3}+{b}^{3}=760$