Solve f(x)=4x^4-17x^2+4

$$f ( x ) = 4 x ^ { 4 } - 17 x ^ { 2 } + 4$$

Factor

$$\left(x-2\right)\left(2x-1\right)\left(x+2\right)\left(2x+1\right)$$

Solution Steps

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

$$4x^{4}-17x^{2}+4=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 4 and q divides the leading coefficient 4. List all candidates \frac{p}{q}.

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

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.

$$x=2$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 4x^{4}-17x^{2}+4 by x-2 to get 4x^{3}+8x^{2}-x-2. To factor the result, solve the equation where it equals to 0.

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

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -2 and q divides the leading coefficient 4. List all candidates \frac{p}{q}.

$$±\frac{1}{2},±1,±2,±\frac{1}{4}$$

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.

$$x=-2$$

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

$$4x^{2}-1=0$$

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, 0 for b, and -1 for c in the quadratic formula.

$$x=\frac{0±\sqrt{0^{2}-4\times 4\left(-1\right)}}{2\times 4}$$

Do the calculations.

$$x=\frac{0±4}{8}$$

Solve the equation 4x^{2}-1=0 when ± is plus and when ± is minus.

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

Rewrite the factored expression using the obtained roots.

$$\left(x-2\right)\left(2x-1\right)\left(x+2\right)\left(2x+1\right)$$

Evaluate

$$4x^{4}-17x^{2}+4$$