Solve x^4+4*3x^3+6*9x^2+4*27x+81

$${ x }^{ 4 } +4 \times 3 { x }^{ 3 } +6 \times 9 { x }^{ 2 } +4 \times 27x+81$$

Factor

$$\left(x+3\right)^{4}$$

Solution Steps

Multiply and combine like terms.

$$x^{4}+12x^{3}+54x^{2}+108x+81$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 81 and q divides the leading coefficient 1. One such root is -3. Factor the polynomial by dividing it by x+3.

$$\left(x+3\right)\left(x^{3}+9x^{2}+27x+27\right)$$

Consider x^{3}+9x^{2}+27x+27. Use the binomial cube formula, a^{3}+3a^{2}b+3ab^{2}+b^{3}=\left(a+b\right)^{3}, where a=x and b=3.

$$\left(x+3\right)^{3}$$

Rewrite the complete factored expression.

$$\left(x+3\right)^{4}$$

Evaluate

$$\left(x+3\right)^{4}$$

Solution Steps

Do the multiplications.

$$x^{4}+12x^{3}+54x^{2}+108x+81$$