Solve frac{3x^{7}}{128}+frac{21x^{6}}{64}+63x^5/32+105x^4/16+105x^3/8+63x^2/4+frac{21x}{2}+3

$$\frac { 3 x ^ { 7 } } { 128 } + \frac { 21 x ^ { 6 } } { 64 } + \frac { 63 x ^ { 5 } } { 32 } + \frac { 105 x ^ { 4 } } { 16 } + \frac { 105 x ^ { 3 } } { 8 } + \frac { 63 x ^ { 2 } } { 4 } + \frac { 21 x } { 2 } + 3$$

Evaluate

$$\frac{3\left(x+2\right)^{7}}{128}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 128 and 64 is 128. Multiply \frac{21x^{6}}{64} times \frac{2}{2}.

$$\frac{3x^{7}}{128}+\frac{2\times 21x^{6}}{128}+\frac{63x^{5}}{32}+\frac{105x^{4}}{16}+\frac{105x^{3}}{8}+\frac{63x^{2}}{4}+\frac{21x}{2}+3$$

Since \frac{3x^{7}}{128} and \frac{2\times 21x^{6}}{128} have the same denominator, add them by adding their numerators.

$$\frac{3x^{7}+2\times 21x^{6}}{128}+\frac{63x^{5}}{32}+\frac{105x^{4}}{16}+\frac{105x^{3}}{8}+\frac{63x^{2}}{4}+\frac{21x}{2}+3$$

Do the multiplications in 3x^{7}+2\times 21x^{6}.

$$\frac{3x^{7}+42x^{6}}{128}+\frac{63x^{5}}{32}+\frac{105x^{4}}{16}+\frac{105x^{3}}{8}+\frac{63x^{2}}{4}+\frac{21x}{2}+3$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 128 and 32 is 128. Multiply \frac{63x^{5}}{32} times \frac{4}{4}.

$$\frac{3x^{7}+42x^{6}}{128}+\frac{4\times 63x^{5}}{128}+\frac{105x^{4}}{16}+\frac{105x^{3}}{8}+\frac{63x^{2}}{4}+\frac{21x}{2}+3$$

Since \frac{3x^{7}+42x^{6}}{128} and \frac{4\times 63x^{5}}{128} have the same denominator, add them by adding their numerators.

$$\frac{3x^{7}+42x^{6}+4\times 63x^{5}}{128}+\frac{105x^{4}}{16}+\frac{105x^{3}}{8}+\frac{63x^{2}}{4}+\frac{21x}{2}+3$$

Do the multiplications in 3x^{7}+42x^{6}+4\times 63x^{5}.

$$\frac{3x^{7}+42x^{6}+252x^{5}}{128}+\frac{105x^{4}}{16}+\frac{105x^{3}}{8}+\frac{63x^{2}}{4}+\frac{21x}{2}+3$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 128 and 16 is 128. Multiply \frac{105x^{4}}{16} times \frac{8}{8}.

$$\frac{3x^{7}+42x^{6}+252x^{5}}{128}+\frac{8\times 105x^{4}}{128}+\frac{105x^{3}}{8}+\frac{63x^{2}}{4}+\frac{21x}{2}+3$$

Since \frac{3x^{7}+42x^{6}+252x^{5}}{128} and \frac{8\times 105x^{4}}{128} have the same denominator, add them by adding their numerators.

$$\frac{3x^{7}+42x^{6}+252x^{5}+8\times 105x^{4}}{128}+\frac{105x^{3}}{8}+\frac{63x^{2}}{4}+\frac{21x}{2}+3$$

Do the multiplications in 3x^{7}+42x^{6}+252x^{5}+8\times 105x^{4}.

$$\frac{3x^{7}+42x^{6}+252x^{5}+840x^{4}}{128}+\frac{105x^{3}}{8}+\frac{63x^{2}}{4}+\frac{21x}{2}+3$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 128 and 8 is 128. Multiply \frac{105x^{3}}{8} times \frac{16}{16}.

$$\frac{3x^{7}+42x^{6}+252x^{5}+840x^{4}}{128}+\frac{16\times 105x^{3}}{128}+\frac{63x^{2}}{4}+\frac{21x}{2}+3$$

Since \frac{3x^{7}+42x^{6}+252x^{5}+840x^{4}}{128} and \frac{16\times 105x^{3}}{128} have the same denominator, add them by adding their numerators.

$$\frac{3x^{7}+42x^{6}+252x^{5}+840x^{4}+16\times 105x^{3}}{128}+\frac{63x^{2}}{4}+\frac{21x}{2}+3$$

Do the multiplications in 3x^{7}+42x^{6}+252x^{5}+840x^{4}+16\times 105x^{3}.

$$\frac{3x^{7}+42x^{6}+252x^{5}+840x^{4}+1680x^{3}}{128}+\frac{63x^{2}}{4}+\frac{21x}{2}+3$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 128 and 4 is 128. Multiply \frac{63x^{2}}{4} times \frac{32}{32}.

$$\frac{3x^{7}+42x^{6}+252x^{5}+840x^{4}+1680x^{3}}{128}+\frac{32\times 63x^{2}}{128}+\frac{21x}{2}+3$$

Since \frac{3x^{7}+42x^{6}+252x^{5}+840x^{4}+1680x^{3}}{128} and \frac{32\times 63x^{2}}{128} have the same denominator, add them by adding their numerators.

$$\frac{3x^{7}+42x^{6}+252x^{5}+840x^{4}+1680x^{3}+32\times 63x^{2}}{128}+\frac{21x}{2}+3$$

Do the multiplications in 3x^{7}+42x^{6}+252x^{5}+840x^{4}+1680x^{3}+32\times 63x^{2}.

$$\frac{3x^{7}+42x^{6}+252x^{5}+840x^{4}+1680x^{3}+2016x^{2}}{128}+\frac{21x}{2}+3$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 128 and 2 is 128. Multiply \frac{21x}{2} times \frac{64}{64}.

$$\frac{3x^{7}+42x^{6}+252x^{5}+840x^{4}+1680x^{3}+2016x^{2}}{128}+\frac{64\times 21x}{128}+3$$

Since \frac{3x^{7}+42x^{6}+252x^{5}+840x^{4}+1680x^{3}+2016x^{2}}{128} and \frac{64\times 21x}{128} have the same denominator, add them by adding their numerators.

$$\frac{3x^{7}+42x^{6}+252x^{5}+840x^{4}+1680x^{3}+2016x^{2}+64\times 21x}{128}+3$$

Do the multiplications in 3x^{7}+42x^{6}+252x^{5}+840x^{4}+1680x^{3}+2016x^{2}+64\times 21x.

$$\frac{3x^{7}+42x^{6}+252x^{5}+840x^{4}+1680x^{3}+2016x^{2}+1344x}{128}+3$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{128}{128}.

$$\frac{3x^{7}+42x^{6}+252x^{5}+840x^{4}+1680x^{3}+2016x^{2}+1344x}{128}+\frac{3\times 128}{128}$$

Since \frac{3x^{7}+42x^{6}+252x^{5}+840x^{4}+1680x^{3}+2016x^{2}+1344x}{128} and \frac{3\times 128}{128} have the same denominator, add them by adding their numerators.

$$\frac{3x^{7}+42x^{6}+252x^{5}+840x^{4}+1680x^{3}+2016x^{2}+1344x+3\times 128}{128}$$

Do the multiplications in 3x^{7}+42x^{6}+252x^{5}+840x^{4}+1680x^{3}+2016x^{2}+1344x+3\times 128.

$$\frac{3x^{7}+42x^{6}+252x^{5}+840x^{4}+1680x^{3}+2016x^{2}+1344x+384}{128}$$

Factor

$$\frac{3\left(x+2\right)^{7}}{128}$$

Solution Steps

Factor out \frac{3}{128}.

$$\frac{3\left(x^{7}+14x^{6}+84x^{5}+280x^{4}+560x^{3}+672x^{2}+448x+128\right)}{128}$$

Consider x^{7}+14x^{6}+84x^{5}+280x^{4}+560x^{3}+672x^{2}+448x+128. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 128 and q divides the leading coefficient 1. One such root is -2. Factor the polynomial by dividing it by x+2.

$$\left(x+2\right)\left(x^{6}+12x^{5}+60x^{4}+160x^{3}+240x^{2}+192x+64\right)$$

Consider x^{6}+12x^{5}+60x^{4}+160x^{3}+240x^{2}+192x+64. 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. One such root is -2. Factor the polynomial by dividing it by x+2.

$$\left(x+2\right)\left(x^{5}+10x^{4}+40x^{3}+80x^{2}+80x+32\right)$$

Consider x^{5}+10x^{4}+40x^{3}+80x^{2}+80x+32. 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. One such root is -2. Factor the polynomial by dividing it by x+2.

$$\left(x+2\right)\left(x^{4}+8x^{3}+24x^{2}+32x+16\right)$$

Consider x^{4}+8x^{3}+24x^{2}+32x+16. 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. One such root is -2. Factor the polynomial by dividing it by x+2.

$$\left(x+2\right)\left(x^{3}+6x^{2}+12x+8\right)$$

Consider x^{3}+6x^{2}+12x+8. Use the binomial cube formula, a^{3}+3a^{2}b+3ab^{2}+b^{3}=\left(a+b\right)^{3}, where a=x and b=2.

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

Rewrite the complete factored expression.

$$\frac{3\left(x+2\right)^{7}}{128}$$