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# Multiplication of Monomials by Polynomials

## Distribute single terms by multiplying them by other terms

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Distributive Property

Learning Goal

I am learning to multiply a monomial and a polynomial using the distributive property.

Warm Up

Recall the formula for the area of a rectangle Area = Length x Width.

[Figure1]

How can we write a simplified expression for the area of this rectangle?

Action

Just as we can add and subtract polynomials, we can also multiply them.

Multiplying a Polynomial by a Monomial

When multiplying polynomials, we must remember the exponent rules. Especially important is the product rule: xnxm=xn+m\begin{align*}x^n \cdot x^m=x^{n+m}\end{align*}.

If the expressions we are multiplying have coefficients and more than one variable, we multiply the coefficients just as we would any number and we apply the product rule on each variable separately.

#### Example A

Multiply the following monomials.

a) (2x2)(5x3)\begin{align*}(2x^2)(5x^3)\end{align*}

b) (3y4)(2y2)\begin{align*}(-3y^4)(2y^2)\end{align*}

Solution

a) (2x2)(5x3)=(25)(x2x3)=10x2+3=10x5\begin{align*}(2x^2)(5x^3)=(2 \cdot 5) \cdot (x^2 \cdot x^3)=10x^{2+3} = 10x^5\end{align*}

b) (3y4)(2y2)=(32)(y4y2)=6y4+2=6y6\begin{align*}(-3y^4)(2y^2)=(-3 \cdot 2) \cdot (y^4 \cdot y^2)=-6y^{4+2}=-6y^6\end{align*}

To multiply a polynomial by a monomial, we have to use the Distributive PropertyThis property says that we need to multiply each term inside the bracket by the term outside the bracket.  a(b+c)=ab+ac\begin{align*}a(b + c) = ab + ac\end{align*}.

#### Example B

Multiply:

a) 3(x2+3x5)\begin{align*}3(x^2+3x-5)\end{align*}

b) 4x(3x27)\begin{align*}4x(3x^2-7)\end{align*}

c) 7y(4y22y+1)\begin{align*}-7y(4y^2-2y+1)\end{align*}

Solution

a) 3(x2+3x5)=3(x2)+3(3x)3(5)=3x2+9x15\begin{align*}3(x^2+3x-5)=3(x^2)+3(3x)-3(5)=3x^2+9x-15\end{align*}

b) 4x(3x27)=(4x)(3x2)+(4x)(7)=12x328x\begin{align*}4x(3x^2-7)=(4x)(3x^2)+(4x)(-7)=12x^3-28x\end{align*}

c)

7y(4y22y+1)=(7y)(4y2)+(7y)(2y)+(7y)(1)=28y3+14y27y\begin{align*}-7y(4y^2-2y+1)&=(-7y)(4y^2)+(-7y)(-2y)+(-7y)(1)\\ &=-28y^3+14y^2-7y\end{align*}

Notice that when we use the Distributive Property, the problem becomes a matter of just multiplying monomials by monomials and adding all the separate parts together.

#### Example C

Multiply 2a2b4(3ab2+7a3b9a+3)\begin{align*}-2a^2b^4(3ab^2+7a^3b-9a+3)\end{align*}.

Solution:

Multiply the monomial by each term inside the parenthesis:

2a2b4(3ab2+7a3b9a+3)=(2a2b4)(3ab2)+(2a2b4)(7a3b)+(2a2b4)(9a)+(2a2b4)(3)=6a3b614a5b5+18a5b46a2b4\begin{align*}& -2a^2b^4(3ab^2+7a^3b-9a+3)\\ & = (-2a^2b^4)(3ab^2)+(-2a^2b^4)(7a^3b)+(-2a^2b^4)(-9a)+(-2a^2b^4)(3)\\ & = -6a^3b^6-14a^5b^5+18a^5b^4-6a^2b^4\end{align*}

We could also use Algebra Tiles!

Lets take a look back at our warm up question. How can we use the distributive property to write a simplified expression for the area of this rectangle?

[Figure2]

Solution:

[Figure3]

Consolidation

Check out this video summarising distributive property

Word Wall

Distributive Property

A method used to multiply or expand a monomial into a polynomial. When using the distributive property multiply each term inside the bracket by the term outside the bracket.  a(b+c)=ab+ac\begin{align*}a(b + c) = ab + ac\end{align*}.

Practice

Multiply the following monomials.

1. (2x)(7x)\begin{align*}(2x)(-7x)\end{align*}
2. (10x)(3xy)\begin{align*}(10x)(3xy)\end{align*}
3. (4mn)(0.5nm2)\begin{align*}(4mn)(0.5nm^2)\end{align*}
4. (5a2b)(12a3b3)\begin{align*}(-5a^2b)(-12a^3b^3)\end{align*}
5. (3xy2z2)(15x2yz3)\begin{align*}(3xy^2z^2)(15x^2yz^3)\end{align*}

Multiply and simplify.

1. 17(8x10)\begin{align*}17(8x-10)\end{align*}
2. 2x(4x5)\begin{align*}2x(4x-5)\end{align*}
3. 9x3(3x22x+7)\begin{align*}9x^3(3x^2-2x+7)\end{align*}
4. 3x(2y2+y5)\begin{align*}3x(2y^2+y-5)\end{align*}
5. 10q(3q2r+5r)\begin{align*}10q(3q^2r+5r)\end{align*}

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