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\left(x^{2}\right)^{3}-12\left(x^{2}\right)^{2}+48x^{2}-64=0
Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(x^{2}-4\right)^{3}.
x^{6}-12\left(x^{2}\right)^{2}+48x^{2}-64=0
To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.
x^{6}-12x^{4}+48x^{2}-64=0
To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.
±64,±32,±16,±8,±4,±2,±1
By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -64 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.
x=2
Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.
x^{5}+2x^{4}-8x^{3}-16x^{2}+16x+32=0
By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{6}-12x^{4}+48x^{2}-64 by x-2 to get x^{5}+2x^{4}-8x^{3}-16x^{2}+16x+32. Solve the equation where the result equals to 0.
±32,±16,±8,±4,±2,±1
By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 32 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.
x=2
Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.
x^{4}+4x^{3}-16x-16=0
By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{5}+2x^{4}-8x^{3}-16x^{2}+16x+32 by x-2 to get x^{4}+4x^{3}-16x-16. Solve the equation where the result equals to 0.
±16,±8,±4,±2,±1
By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -16 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.
x=2
Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.
x^{3}+6x^{2}+12x+8=0
By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{4}+4x^{3}-16x-16 by x-2 to get x^{3}+6x^{2}+12x+8. Solve the equation where the result equals to 0.
±8,±4,±2,±1
By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 8 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.
x=-2
Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.
x^{2}+4x+4=0
By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}+6x^{2}+12x+8 by x+2 to get x^{2}+4x+4. Solve the equation where the result equals to 0.
x=\frac{-4±\sqrt{4^{2}-4\times 1\times 4}}{2}
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, 4 for b, and 4 for c in the quadratic formula.
x=\frac{-4±0}{2}
Do the calculations.
x=-2
Solutions are the same.
x=2 x=-2
List all found solutions.