Wize High School Grade 9 Math Textbook > Polynomial Expressions

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(2x)

=(3×2)x=(3\times2)x=(3×2)x

=6x=6x=6x

b) −2(−5x2)-2\left(-5x^2\right)−2(−5x2)

=(−2×−5)x2=(-2\times-5)x^2=(−2×−5)x2

=10x2=10x^2=10x2

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

=(−6×12)xy=\left(-6\times\dfrac{1}{2}\right)xy=(−6×21​)xy

=−3xy=-3xy=−3xy

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Multiplying a monomial and a monomial
Simplify the following:

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

(3×2)(x×x)(3\times2)(x\times x)(3×2)(x×x)

=6x2=6x^2=6x2

b) (−2x)(−5x2)(-2x)\left(-5x^2\right)(−2x)(−5x2)

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

=10x3=10x^3=10x3

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

=(7×4)(x3×y)=(7\times 4)(x^3\times y)=(7×4)(x3×y)

=28x3y=28x^3y=28x3y

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

=(−6×12)(y3×xy)=\left(-6\times\dfrac{1}{2}\right)(y^3\times xy)=(−6×21​)(y3×xy)

=−3xy4=-3xy^4=−3xy4

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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+3)

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

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

b) (4x2−3x)(−2x)(4x^2-3x)(-2x)(4x2−3x)(−2x)

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

=−8x3+6x2=-8x^3+6x^2=−8x3+6x2

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

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

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

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Example: Multiplying a Polnomial & Monomial
Expand and simplify the following.

a) 4(2x−3)4(2x-3)4(2x−3)

4(2x−3)\bct4(2x-3)4(2x−3)

=4(2x)+4(−3)=\bct4(2x)+\bct4(-3)=4(2x)+4(−3)

=8x−12=8x-12=8x−12

b) 5x(2−x+3x2)5x(2-x+3x^2)5x(2−x+3x2)

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

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

=10x−5x2+15x3=10x-5x^2+15x^3=10x−5x2+15x3

c) (3n2−4)(−2n)(3n^2-4)(-2n)(3n2−4)(−2n)

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

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

=−6n3+8n=-6n^3+8n=−6n3+8n

d) −(3x2−2xy−4y+4)-(3x^2-2xy-4y+4)−(3x2−2xy−4y+4)

=−3x2+2xy+4y−4=-3x^2+2xy+4y-4=−3x2+2xy+4y−4

Practice: Multiplying a Polynomial & Monomial
Expand and simplify −3x(4x2+2x−5)-3x(4x^2+2x-5)−3x(4x2+2x−5).

Answer

I don't know

Practice: Multiplying a Polynomial & Monomial
Fill in the missing blanks:

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

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

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

\boxed{A}=

\boxed{B}=

\boxed{C}=

I don't know

Practice: Multiplying a Polynomial & Monomial
Write a simplified expression for the perimeter and area for each of the following shapes.

Perimeter =

Area =

I don't know