Solve a^5-6a^4+16a^3-32a^2+48a-32

$$a ^ { 5 } - 6 a ^ { 4 } + 16 a ^ { 3 } - 32 a ^ { 2 } + 48 a - 32$$

Factor

$$\left(a^{2}+4\right)\left(a-2\right)^{3}$$

Solution Steps

To factor the expression, solve the equation where it equals to 0.

$$a^{5}-6a^{4}+16a^{3}-32a^{2}+48a-32=0$$

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}.

$$±32,±16,±8,±4,±2,±1$$

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.

$$a=2$$

By Factor theorem, a-k is a factor of the polynomial for each root k. Divide a^{5}-6a^{4}+16a^{3}-32a^{2}+48a-32 by a-2 to get a^{4}-4a^{3}+8a^{2}-16a+16. To factor the result, solve the equation where it equals to 0.

$$a^{4}-4a^{3}+8a^{2}-16a+16=0$$

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}.

$$±16,±8,±4,±2,±1$$

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.

$$a=2$$

By Factor theorem, a-k is a factor of the polynomial for each root k. Divide a^{4}-4a^{3}+8a^{2}-16a+16 by a-2 to get a^{3}-2a^{2}+4a-8. To factor the result, solve the equation where it equals to 0.

$$a^{3}-2a^{2}+4a-8=0$$

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}.

$$±8,±4,±2,±1$$

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.

$$a=2$$

By Factor theorem, a-k is a factor of the polynomial for each root k. Divide a^{3}-2a^{2}+4a-8 by a-2 to get a^{2}+4. To factor the result, solve the equation where it equals to 0.

$$a^{2}+4=0$$

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, 0 for b, and 4 for c in the quadratic formula.

$$a=\frac{0±\sqrt{0^{2}-4\times 1\times 4}}{2}$$

Do the calculations.

$$a=\frac{0±\sqrt{-16}}{2}$$

Polynomial a^{2}+4 is not factored since it does not have any rational roots.

$$a^{2}+4$$

Rewrite the factored expression using the obtained roots.

$$\left(a^{2}+4\right)\left(a-2\right)^{3}$$

Evaluate

$$\left(a^{2}+4\right)\left(a-2\right)^{3}$$