Multiplying Polynomials by Monomial

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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`.