Solve (3/2+x5/3x^2+1/2x^4)(3/4+frac{6}{4}x+frac{3}{3}x^2)

$$( \frac { 3 } { 2 } + x \frac { 5 } { 3 } x ^ { 2 } + \frac { 1 } { 2 } x ^ { 4 } ) ( \frac { 3 } { 4 } + \frac { 6 } { 4 } x + \frac { 3 } { 3 } x ^ { 2 } )$$

Evaluate

$$\frac{\left(4x^{2}+6x+3\right)\left(3x^{4}+10x^{3}+9\right)}{24}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$\left(\frac{3}{2}+x^{3}\times \frac{5}{3}+\frac{1}{2}x^{4}\right)\left(\frac{3}{4}+\frac{6}{4}x+\frac{3}{3}x^{2}\right)$$

Divide 3 by 3 to get 1.

$$\left(\frac{3}{2}+x^{3}\times \frac{5}{3}+\frac{1}{2}x^{4}\right)\left(\frac{3}{4}+\frac{6}{4}x+1x^{2}\right)$$

Reduce the fraction \frac{6}{4} to lowest terms by extracting and canceling out 2.

$$\left(\frac{3}{2}+x^{3}\times \frac{5}{3}+\frac{1}{2}x^{4}\right)\left(\frac{3}{4}+\frac{3}{2}x+1x^{2}\right)$$

Use the distributive property to multiply \frac{3}{2}+x^{3}\times \frac{5}{3}+\frac{1}{2}x^{4} by \frac{3}{4}+\frac{3}{2}x+1x^{2} and combine like terms.

$$\frac{9}{8}+\frac{9}{4}x+\frac{3}{2}x^{2}+\frac{3}{4}x^{3}\times \frac{5}{3}+\frac{23}{8}x^{4}+\frac{29}{12}x^{5}+\frac{1}{2}x^{6}$$

Multiply \frac{3}{4} and \frac{5}{3} to get \frac{5}{4}.

$$\frac{9}{8}+\frac{9}{4}x+\frac{3}{2}x^{2}+\frac{5}{4}x^{3}+\frac{23}{8}x^{4}+\frac{29}{12}x^{5}+\frac{1}{2}x^{6}$$

Expand

$$\frac{x^{6}}{2}+\frac{29x^{5}}{12}+\frac{23x^{4}}{8}+\frac{5x^{3}}{4}+\frac{3x^{2}}{2}+\frac{9x}{4}+\frac{9}{8}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$\left(\frac{3}{2}+x^{3}\times \frac{5}{3}+\frac{1}{2}x^{4}\right)\left(\frac{3}{4}+\frac{6}{4}x+\frac{3}{3}x^{2}\right)$$

Divide 3 by 3 to get 1.

$$\left(\frac{3}{2}+x^{3}\times \frac{5}{3}+\frac{1}{2}x^{4}\right)\left(\frac{3}{4}+\frac{6}{4}x+1x^{2}\right)$$

Reduce the fraction \frac{6}{4} to lowest terms by extracting and canceling out 2.

$$\left(\frac{3}{2}+x^{3}\times \frac{5}{3}+\frac{1}{2}x^{4}\right)\left(\frac{3}{4}+\frac{3}{2}x+1x^{2}\right)$$

Use the distributive property to multiply \frac{3}{2}+x^{3}\times \frac{5}{3}+\frac{1}{2}x^{4} by \frac{3}{4}+\frac{3}{2}x+1x^{2} and combine like terms.

$$\frac{9}{8}+\frac{9}{4}x+\frac{3}{2}x^{2}+\frac{3}{4}x^{3}\times \frac{5}{3}+\frac{23}{8}x^{4}+\frac{29}{12}x^{5}+\frac{1}{2}x^{6}$$

Multiply \frac{3}{4} and \frac{5}{3} to get \frac{5}{4}.

$$\frac{9}{8}+\frac{9}{4}x+\frac{3}{2}x^{2}+\frac{5}{4}x^{3}+\frac{23}{8}x^{4}+\frac{29}{12}x^{5}+\frac{1}{2}x^{6}$$