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Wize High School Algebra I Textbook (Common Core) > Polynomial Expressions

Monomial ×\times Polynomial (Distributive Method)

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Multiplying a Polynomial by a Monomial

When adding or subtracting polynomials, we can only add or subtract like terms (where the variable part of the terms are the exact same).



When multiplying polynomials, we can multiply any terms together, they don't have to be like terms!
  • We multiply the coefficients together (the number parts)
  • We multiply the variables together (remember to use the exponent product rule for multiplying powers)

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Multiplying a constant and a monomial

Exapnd the following: (i.e. simplify)

a) 3(2x)3\left(2x\right)

=(3×2)x=(3\times2)x

=6x=6x

b) −2(−5x2)-2\left(-5x^2\right)

=(−2×−5)x2=(-2\times-5)x^2

=10x2=10x^2

c) −6(xy2)-6\left(\dfrac{xy}{2}\right)

=(−6×12)xy=\left(-6\times\dfrac{1}{2}\right)xy

=−3xy=-3xy
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Multiplying a monomial and a monomial

Simplify the following:

a) (3x)(2x)(3x)\left(2x\right)

(3×2)(x×x)(3\times2)(x\times x)

=6x2=6x^2

b) (−2x)(−5x2)(-2x)\left(-5x^2\right)

(−2×−5)(x×x2)(-2\times-5)(x\times x^2)

=10x3=10x^3

c) (7x3)(4y)(7x^3)(4y)

=(7×4)(x3×y)=(7\times 4)(x^3\times y)

=28x3y=28x^3y

d) (−6y3)(xy2)(-6y^3)\left(\dfrac{xy}{2}\right)

=(−6×12)(y3×xy)=\left(-6\times\dfrac{1}{2}\right)(y^3\times xy)

=−3xy4=-3xy^4
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Multiplying a monomial and a polynomial (Distributive Property)

We follow the distributive property -- multiply each term in the polynomial by the monomial.



Simplify the following:

a) 2x(x+3)2x(x+3)

=2x(x)+2x(3)=2x(x)+2x(3)

=2x2+6x=2x^2+6x

b) (4x2−3x)(−2x)(4x^2-3x)(-2x)

=(4x2)(−2x)+(−3x)(−2x)=(4x^2)(-2x)+(-3x)(-2x)

=−8x3+6x2=-8x^3+6x^2

c) −5xy2(x3−2xy+4)-5xy^2(x^3-2xy+4)

=(−5xy2)(x3)+(−5xy2)(−2xy)+(−5xy2)(4)=(-5xy^2)(x^3)+(-5xy^2)(-2xy)+(-5xy^2)(4)

=−5x4y2+10x2y3−20xy2=-5x^4y^2+10x^2y^3-20xy^2

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Example: Multiplying a Polnomial & Monomial

Expand and simplify the following.

a) 4(2x−3)4(2x-3)

4(2x−3)\bct4(2x-3)

=4(2x)+4(−3)=\bct4(2x)+\bct4(-3)

=8x−12=8x-12

b) 5x(2−x+3x2)5x(2-x+3x^2)

5x(2−x+3x2)\bct{5x}(2-x+3x^2)

=5x(2)+5x(−x)+5x(3x2)=\bct{5x}(2)+\bct{5x}(-x)+\bct{5x}(3x^2)

=10x−5x2+15x3=10x-5x^2+15x^3

c) (3n2−4)(−2n)(3n^2-4)(-2n)

=(3n2−4)(−2n)=(3n^2-4)(\bct{-2n})

=3n2(−2n)−4(−2n)=3n^2(\bct{-2n})-4(\bct{-2n})

=−6n3+8n=-6n^3+8n

d) −(3x2−2xy−4y+4)-(3x^2-2xy-4y+4)

=−3x2+2xy+4y−4=-3x^2+2xy+4y-4

Practice: Multiplying a Polynomial & Monomial

Expand and simplify −3x(4x2+2x−5)-3x(4x^2+2x-5).

Practice: Simplifying Polynomial Expressions

Expand and simplify the following:

a) 3x(x2−x+1)+x(4−x)3x(x^2-x+1)+x(4-x)

b) 12(45y−213)−25(12−y2)\dfrac{1}{2}\left(\dfrac{4}{5}y-2 \dfrac{1}{3}\right)-\dfrac{2}{5}\left(\dfrac{1}{2}-\dfrac{y}{2}\right)

Practice: Simplifying Polynomial Expressions

Simplify 2x(4−x+3x2)−y(y+2y2−1)2x(4-x+3x^2)-y(y+2y^2-1) and evaluate the expression when x=1x=1 and y=−2y=-2.

Practice: Multiplying a Polynomial & Monomial

Fill in the missing blanks:

a) 2x(A−3)=−8x2−6x2x\left(\boxed{A}-3\right)=-8x^2-6x

b) B(2t2+t−1)=10t3+5t2−5t\boxed{B}(2t^2+t-1)=10t^3+5t^2-5t

c) −6x(4x2+C−2y)=−24x3+8x2y+12xy-6x\left(4x^2+\boxed{C}-2y\right)=-24x^3+8x^2y+12xy

Practice: Multiplying a Polynomial & Monomial

Write a simplified expression for the perimeter and area for each of the following shapes.


Practice: Simplifying Polynomial Expressions

A gardening company is designing a new backyard and has created the following blueprint.

a) Create an expression representing the area of the pool.

b) Create an expression representing the area of the patio.

c) Create an expression representing the area of the field.

d) Create an expression representing the area of the entire backyard.

e) Find the area of the entire backyard when x=2x=2