# Multiplying Polynomials by Monomial

## Related Calculator: Polynomial Calculator

To multiply polynomial by monomial, one should use distributive property of multiplication.

Then, just multiply monomials and you're done.

Example 1. Multiply (2x)(3x^2+5x+4).

color(red)((2x))color(green)((3x^2+5x+4))=

=color(red)((2x))*color(green)((3x^2))+color(red)((2x))*color(green)((5x))+color(red)((2x))*color(green)((4))= (distributive property of multiplication)

=6x^3+10x^2+8x (multiply monomials)

Answer: (2x)(3x^2+5x+4)=6x^3+10x^2+8x.

Negative terms are handled in the same way.

Example 2. Multiply the following: (x^3-5x^2-x+7)1/3x^2.

color(green)((x^3-5x^2-x+7))color(red)(1/3x^2)=

=color(green)(x^3)*color(red)(1/3x^2)+color(green)((-5x^2))*color(red)(1/3x^2)+color(green)((-x))*color(red)(1/3x^2)+color(green)(7)*color(red)(1/3x^2)= (distributive property of multiplication)

=1/3x^5-5/3x^4-1/3x^3+7/3x^2 (multiply monomials)

Answer: (x^3-5x^2-x)1/3x^2=1/3x^5-5/3x^4-1/3x^3+7/3x^2.

Of course, polynomials with many variables can also be handled in a similar way.

Example 3. Multiply -3xy^2 by (3x^2y+2xz-5xy^2-z).

-3xy^2(3x^2y+2xz-5xy^2-z)=

=(-3xy^2)(3x^2y)+(-3xy^2)(2xz)+(-3xy^2)(-5xy^2)+(-3xy^2)(-z)=

=-9x^3y^3-6x^2y^2z+15x^2y^4+3xy^2z.

Answer: -3xy^2(3x^2y+2xz-5xy^2-z)=-9x^3y^3-6x^2y^2z+15x^2y^4+3xy^2z.

Now, it is time to exercise.

Exercise 1. Multiply (x^3+2x+4)*(5x^2).

Answer: 5x^5+10x^3+20x^2.

Exercise 2. Multiply -2/7a^3(a^3-2a^2+7b).

Answer: -2/7a^6+4/7a^5-2a^3b.

Exercise 3. Multiply (-3ab)(5a^2b-3a^3bc+3/5a^2b^2-1/10ab).

Answer: -15a^3b^2+9a^4b^2c-9/5a^3b^3+3/10a^2b^2.