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Wize High School Grade 11 Math Textbook > Operations with Polynomials

Multiplying Any Polynomials

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Binomial x Trinomial

It turns out that the distributive method ("hand-shake" rule) can be used to multiply any two polynomials together, including a binomial ×\times a trinomial!

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


Begin by taking each term in the first polynomial and distributing it to each term in the second:

=(3x−1)(2x2−4x+1)=(\bcth{3x}\bct{-1})(2x^2-4x+1)

=(3x)(2x2)+(3x)(−4x)+(3x)(+1)        +        (−1)(2x2)+(−1)(−4x)+(−1)(+1)=(\bcth{3x})(2x^2)+(\bcth{3x})(-4x)+(\bcth{3x})(+1)~~~~~~~~+~~~~~~~~(\bct{-1})(2x^2)+(\bct{-1})(-4x)+(\bct{-1})(+1)

=     6x3          −12x2          +3x                            −2x2               +4x               −1=~~~~~6x^3~~~~~~~~~~-12x^2~~~~~~~~~~+3x~~~~~~~~~~~~~~~~~~~~~~~~~~~~-2x^2~~~~~~~~~~~~~~~+4x~~~~~~~~~~~~~~~-1

Now we simplify our answer by combining any like terms:

=     6x3−14x2+7x−1=~~~~~6x^3-14x^2+7x-1
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Example: Multiplying Binomials & Trinomials

Expand and simplify the given expression.

(2k+4)(3k2−6k+5)\left(2k+4\right)\left(3k^2-6k+5\right)

         (2k+4)(3k2−6k+5)~~~~~~~~~(\bct{2k}\bcth{+4})(3k^2-6k+5)


=(2k)(3k2)+(2k)(−6k)+(2k)(+5)+(+4)(3k2)+(+4)(−6k)+(+4)(+5)=6k3−12k2+10k+12k2−24k+20=6k3+0k2−14k+20\begin{array}{rccccccccccc} =&(\bct{2k})(3k^2)&+&(\bct{2k})(-6k)&+&(\bct{2k})(+5)&+&(\bcth{+4})(3k^2)&+&(\bcth{+4})(-6k)&+&(\bcth{+4})(+5)\\ =&\bct{6k^3}&&\bct{-12k^2}&&\bct{+10k}&&\bcth{+12k^2}&&\bcth{-24k}&&\bcth{+20}\\ =&6k^3&&+0k^2&&-14k&&+20 \end{array}

So, this simplifies to 6k3−14k+20\boxed{6k^3-14k+20}.

Practice: Multiplying Binomials X Trinomials

Expand and simplify.

(2y+1)(5y2−3y+10)\left(2y+1\right)\left(5y^2-3y+10\right)

Practice: Multiply Polynomials

Use any method to multiply the expression.
(x−2)(x2+x)(3x−7)(x-2)(x^2 + x)(3x - 7)

Choose the answer that represents the simplified polynomial.


Practice: Multiplying Polynomials

The formula for the surface area of a rectangular solid is given by the formula: S=2ab+2bc+2acS=2ab + 2bc + 2ac

Find a function that describes the surface area in terms of aa given that
  • The length of bb is 3 more than aa
  • The length of cc is 1 less than aa

Practice: Multiplying Polynomials

Find an expression to represent the area of the following parallelogram. Enter your answer in expanded and simplified form.


Practice: Multiplying Polynomials

You and your friends are building a tree-house and need to purchase some 2' by 4' lumber. The lumber cost 0.1a−120.1a-12 each, and the amount of lumber needed is represented by 70+200a−a270+200a-a^2, where aa represents the size of tree-house, measured in square feet.

a) Write a simplified expression in the variable aa to represent the total cost of lumber needed for this tree-house project.

b) What is the total cost of lumber of lumber required to build a 200 ft2 tree-house?