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Zorluk: Çok zorEquivalent Algebraic Expressions

For all x>0x > 0, the expression (x4/3+4x2/3+16x2/3+2x1/3+4+2x1/3)3−x2−12x4/3−48x2/3\left( \frac{x^{4/3} + 4x^{2/3} + 16}{x^{2/3} + 2x^{1/3} + 4} + 2x^{1/3} \right)^3 - x^2 - 12x^{4/3} - 48x^{2/3} is equivalent to a constant CC. What is the value of CC?

Cevap: 64

Cevap

The constant value is 64.
The expression inside the parentheses simplifies to x2/3+4x^{2/3} + 4 after factoring the numerator as (x2/3+2x1/3+4)(x2/3−2x1/3+4)(x^{2/3} + 2x^{1/3} + 4)(x^{2/3} - 2x^{1/3} + 4) and canceling the common factor in the denominator. Cubing x2/3+4x^{2/3} + 4 yields x2+12x4/3+48x2/3+64x^2 + 12x^{4/3} + 48x^{2/3} + 64. Subtracting the remaining terms x2+12x4/3+48x2/3x^2 + 12x^{4/3} + 48x^{2/3} from this expansion results in the constant value 64.

Adım Adım Çözüm

1
Substitute u=x1/3u = x^{1/3} into the expression to simplify the fractional exponents.
The terms become x1/3=ux^{1/3} = u, x2/3=u2x^{2/3} = u^2, x4/3=u4x^{4/3} = u^4, and x2=u6x^2 = u^6. The expression inside the parentheses is rewritten as u4+4u2+16u2+2u+4+2u\frac{u^4 + 4u^2 + 16}{u^2 + 2u + 4} + 2u.
Using a temporary variable uu simplifies the algebraic factoring and manipulation of terms with fractional exponents.
2
Factor the numerator u4+4u2+16u^4 + 4u^2 + 16 by completing the square.
u4+4u2+16=(u2+4)2−4u2=(u2+2u+4)(u2−2u+4)u^4 + 4u^2 + 16 = (u^2 + 4)^2 - 4u^2 = (u^2 + 2u + 4)(u^2 - 2u + 4).
Expressing the quartic polynomial as a difference of squares allows it to be factored into two quadratic polynomials.
3
Simplify the rational expression and add 2u2u.
(u2+2u+4)(u2−2u+4)u2+2u+4+2u=(u2−2u+4)+2u=u2+4\frac{(u^2 + 2u + 4)(u^2 - 2u + 4)}{u^2 + 2u + 4} + 2u = (u^2 - 2u + 4) + 2u = u^2 + 4.
Canceling the common factor u2+2u+4u^2 + 2u + 4 in the numerator and denominator simplifies the expression inside the parentheses to u2+4u^2 + 4.
4
Substitute u=x1/3u = x^{1/3} back into u2+4u^2 + 4 and cube the expression.
(x2/3+4)3=(x2/3)3+3(x2/3)2(4)+3(x2/3)(16)+64=x2+12x4/3+48x2/3+64(x^{2/3} + 4)^3 = (x^{2/3})^3 + 3(x^{2/3})^2(4) + 3(x^{2/3})(16) + 64 = x^2 + 12x^{4/3} + 48x^{2/3} + 64.
Applying the binomial expansion formula (A+B)3=A3+3A2B+3AB2+B3(A + B)^3 = A^3 + 3A^2B + 3AB^2 + B^3 expands the cubed expression.
5
Subtract the remaining terms from the expanded expression.
(x2+12x4/3+48x2/3+64)−x2−12x4/3−48x2/3=64(x^2 + 12x^{4/3} + 48x^{2/3} + 64) - x^2 - 12x^{4/3} - 48x^{2/3} = 64.
Subtracting the variable terms cancels them out entirely, leaving the constant value 64.

Anahtar Kavram

Factoring quartic polynomials using the difference of squares and simplifying rational expressions with fractional exponents.
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Konular

Equivalent Algebraic Expressions

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