Algebra - Study Mode

[#481] A = $$frac{{{x^8} - 1}}{{{x^4} + 1}}$$ xa0and B = $$frac{{{y^4} - 1}}{{{y^2} + 1}}.$$ xa0If x = 2 and y = 9, then what is the value of A 2 + 2AB + AB 2 ?
Correct Answer

(B) 98625

Explanation

Solution: $$eqalign{
& A = frac{{{x^8} - 1}}{{{x^4} + 1}} = frac{{left( {{x^4} + 1}
ight)left( {{x^4} - 1}
ight)}}{{left( {{x^4} + 1}
ight)}} = {x^4} - 1 cr
& B = frac{{{y^4} - 1}}{{{y^2} + 1}} = frac{{left( {{y^2} + 1}
ight)left( {{y^2} - 1}
ight)}}{{left( {{y^2} + 1}
ight)}} = {y^2} - 1 cr
& x = 2,,,y = 9 cr
& A = {x^4} - 1 = {left( 2
ight)^4} - 1 = 15 cr
& B = {y^2} - 1 = {left( 9
ight)^2} - 1 = 80 cr
& {A^2} + 2AB + A{B^2} cr
& = {left( {15}
ight)^2} + 2 imes 15 imes 80 + 15 imes {left( {80}
ight)^2} cr
& = 15left( {15 + 160 + 6400}
ight) cr
& = 15 imes left( {6575}
ight) cr
& = 98625 cr} $$

[#482] If x 2 - 12x + 33 = 0, then what is the value of (x - 4) 2 + $$frac{1}{{{{left( {x - 4}
ight)}^2}}}?$$
Correct Answer

(B) 14

Explanation

Solution: $$eqalign{
& {x^2} - 12x + 33 = 0 cr
& {x^2} - 12x + 36 - 3 = 0 cr
& {left( {x - 6}
ight)^2} - 3 = 0 cr
& x = sqrt 3 + 6 cr
& {left( {x - 4}
ight)^2} + frac{1}{{{{left( {x - 4}
ight)}^2}}} cr
& herefore {left( {sqrt 3 + 2}
ight)^2} + frac{1}{{{{left( {sqrt 3 + 2}
ight)}^2}}} cr
& = {left( {sqrt 3 + 2}
ight)^2} + {left( {sqrt 3 - 2}
ight)^2} cr
& = 2left( {{{sqrt 3 }^2} + {2^2}}
ight) cr
& = 2 imes 7 cr
& = 14 cr} $$

[#483] If a + b = 8 and a + a 2 b + b + ab 2 = 128 then the positive value of a 3 + b 3 is:
Correct Answer

(D) 152

Explanation

Solution: a + b = 8 a + a 2 b + b + ab 2 = 128 (a + b) + ab(a + b) = 128 (1 + ab)(a + b) = 128 1 + ab = 16 ab = 15 a 3 + b 3 = (a + b) 3 - 3ab(a + b) = 512 - 3 × 15 × 8 = 512 - 360 = 152

[#484] If x 2 + 16 = -4x, then what is the value of x 3 - 64?
Correct Answer

(B) 0

Explanation

Solution: x 2 + 16 = -4x x 3 - 64 = x 3 - (4) 3 = (x - 4)(x 2 + 16 + 4x) = (x - 4)(-4x + 4x) = 0

[#485] If x + y = 1, then what is the value of x 3 + 3xy + y 3 ?
Correct Answer

(B) 1

Explanation

Solution: x + y = 1 put y = 0 x = 1 x 3 + 3xy + y 3 = 1 3 + 0 + 0 = 1