Multiplying Polynomials Calculator

The calculator will multiply two polynomials (quadratic, binomial, trinomial etc.), with steps shown.

First polynomial:

Second polynomial:

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Solution

Your input: multiply $$$3 x^{8} - 5 x^{3} - x^{2} + 2 x + 1$$$ by $$$2 x^{2} - 5 x + 3$$$.

To multiply polynomials, multiply each term of the first polynomial by every term of the second polynomial. Then simplify the products and add them. Finally, simplify further if possible.

So, perform the first step:

$$$\left(\color{DarkCyan}{3 x^{8}}\color{Peru}{- 5 x^{3}}\color{Blue}{- x^{2}}+\color{Purple}{2 x}+\color{DarkBlue}{1}\right) \cdot \left(\color{Green}{2 x^{2}}\color{OrangeRed}{- 5 x}+\color{Magenta}{3}\right)=$$$

$$$=\left(\color{DarkCyan}{3 x^{8}}\right)\cdot \left(\color{Green}{2 x^{2}}\right)+\left(\color{DarkCyan}{3 x^{8}}\right)\cdot \left(\color{OrangeRed}{- 5 x}\right)+\left(\color{DarkCyan}{3 x^{8}}\right)\cdot \left(\color{Magenta}{3}\right)+$$$

$$$+\left(\color{Peru}{- 5 x^{3}}\right)\cdot \left(\color{Green}{2 x^{2}}\right)+\left(\color{Peru}{- 5 x^{3}}\right)\cdot \left(\color{OrangeRed}{- 5 x}\right)+\left(\color{Peru}{- 5 x^{3}}\right)\cdot \left(\color{Magenta}{3}\right)+$$$

$$$+\left(\color{Blue}{- x^{2}}\right)\cdot \left(\color{Green}{2 x^{2}}\right)+\left(\color{Blue}{- x^{2}}\right)\cdot \left(\color{OrangeRed}{- 5 x}\right)+\left(\color{Blue}{- x^{2}}\right)\cdot \left(\color{Magenta}{3}\right)+$$$

$$$+\left(\color{Purple}{2 x}\right)\cdot \left(\color{Green}{2 x^{2}}\right)+\left(\color{Purple}{2 x}\right)\cdot \left(\color{OrangeRed}{- 5 x}\right)+\left(\color{Purple}{2 x}\right)\cdot \left(\color{Magenta}{3}\right)+$$$

$$$+\left(\color{DarkBlue}{1}\right)\cdot \left(\color{Green}{2 x^{2}}\right)+\left(\color{DarkBlue}{1}\right)\cdot \left(\color{OrangeRed}{- 5 x}\right)+\left(\color{DarkBlue}{1}\right)\cdot \left(\color{Magenta}{3}\right)=$$$

Simplify the products:

$$$=6 x^{10}- 15 x^{9}+9 x^{8}+$$$

$$$- 10 x^{5}+25 x^{4}- 15 x^{3}+$$$

$$$- 2 x^{4}+5 x^{3}- 3 x^{2}+$$$

$$$+4 x^{3}- 10 x^{2}+6 x+$$$

$$$+2 x^{2}- 5 x+3=$$$

Simplify further:

$$$=6 x^{10} - 15 x^{9} + 9 x^{8} - 10 x^{5} + 23 x^{4} - 6 x^{3} - 11 x^{2} + x + 3$$$

Answer: $$$\left(3 x^{8} - 5 x^{3} - x^{2} + 2 x + 1\right)\cdot \left(2 x^{2} - 5 x + 3\right)=6 x^{10} - 15 x^{9} + 9 x^{8} - 10 x^{5} + 23 x^{4} - 6 x^{3} - 11 x^{2} + x + 3$$$.