Solve 16m^5-8m^3+m-16m^4-1+8m^2

$$16 m ^ { 5 } - 8 m ^ { 3 } + m - 16 m ^ { 4 } - 1 + 8 m ^ { 2 }$$

Factor

$$\left(m-1\right)\left(4m^{2}-1\right)^{2}$$

Solution Steps

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

$$16m^{5}-16m^{4}-8m^{3}+8m^{2}+m-1=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -1 and q divides the leading coefficient 16. List all candidates \frac{p}{q}.

$$±\frac{1}{16},±\frac{1}{8},±\frac{1}{4},±\frac{1}{2},±1$$

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.

$$m=1$$

By Factor theorem, m-k is a factor of the polynomial for each root k. Divide 16m^{5}-16m^{4}-8m^{3}+8m^{2}+m-1 by m-1 to get 16m^{4}-8m^{2}+1. To factor the result, solve the equation where it equals to 0.

$$16m^{4}-8m^{2}+1=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 1 and q divides the leading coefficient 16. List all candidates \frac{p}{q}.

$$±\frac{1}{16},±\frac{1}{8},±\frac{1}{4},±\frac{1}{2},±1$$

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.

$$m=\frac{1}{2}$$

By Factor theorem, m-k is a factor of the polynomial for each root k. Divide 16m^{4}-8m^{2}+1 by 2\left(m-\frac{1}{2}\right)=2m-1 to get 8m^{3}+4m^{2}-2m-1. To factor the result, solve the equation where it equals to 0.

$$8m^{3}+4m^{2}-2m-1=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -1 and q divides the leading coefficient 8. List all candidates \frac{p}{q}.

$$±\frac{1}{8},±\frac{1}{4},±\frac{1}{2},±1$$

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.

$$m=\frac{1}{2}$$

By Factor theorem, m-k is a factor of the polynomial for each root k. Divide 8m^{3}+4m^{2}-2m-1 by 2\left(m-\frac{1}{2}\right)=2m-1 to get 4m^{2}+4m+1. To factor the result, solve the equation where it equals to 0.

$$4m^{2}+4m+1=0$$

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 4 for a, 4 for b, and 1 for c in the quadratic formula.

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

Do the calculations.

$$m=\frac{-4±0}{8}$$

Solutions are the same.

$$m=-\frac{1}{2}$$

Rewrite the factored expression using the obtained roots.

$$\left(m-1\right)\left(2m-1\right)^{2}\left(2m+1\right)^{2}$$

Evaluate

$$\left(m-1\right)\left(4m^{2}-1\right)^{2}$$