Multiply Polynomials Grade 9

Grade 9 examples on multiplication of polynomials are presented along with their detailed solutions. More questions and their solutions are also included.

Multiply Monomials by Polynomials with Examples

2) mutliply monomials,
3) and add like terms of a plynomial,
\( \) \( \) \( \) \( \) \( \) Example 1
Multiply the following monomials and polynomials
a)   \( 2 (6 x + 2) \quad \) b) \( \quad - 3 x (2 x^2 - x) \quad \)
c) \( \quad -\dfrac{1}{2} x^2 ( 4 x^2 - 2x + 6 x y) \)
Solution to Example 1
a)
Given \( \qquad 2 (6 x + 2) \)
Use the distributive law   \( \color{red}{a} \color{blue}{(b+c)} = \color{red}a \color{blue}b + \color{red}a \color{blue}c \)   to write the given product as the sum of products of monomials
\( \qquad \qquad \color{red}{2} \color{blue}{(6 x + 2)} = \color{red}{2}\color{blue}{(6x)} + \color{red}{2} \color{blue}{(2)} \)
Mulitply constants together and variables together
\( \qquad \qquad = 2(6)(x) + 2(2) \)
Simplify
\( \qquad \qquad = 12 x + 4 \)

b)
Given \( \qquad - 3 x (2 x^2 - x) \)
Use the distributive law   \( \color{red}{a} \color{blue}{(b+c)} = \color{red}a \color{blue}b + \color{red}a \color{blue}c \)   to write the given product as the sum of products of monomials
\( \qquad \qquad \color{red}{- 3 x } \color{blue}{(2 x^2 - x)} = \color{red}{-3x}\color{blue}{(2x^2)} \color{red}{-3x} \color{blue}{(-x)} \)
Mulitply constants together and variables together
\( \qquad \qquad = -3(2)(x x^2) -3(-1)(x x) \)
Simplify
\( \qquad \qquad = -6x^3 + 3x^2 \)

c)
Given \( \qquad -\dfrac{1}{2} x^2 ( 4 x^2 - 2x + 6 x y) \)
Use the distributive law   \( \color{red}{a} \color{blue}{(b+c)} = \color{red}a \color{blue}b + \color{red}a \color{blue}c \)   to write the given product as the sum of products of monomials
\( \qquad \qquad \color{red}{-\dfrac{1}{2} x^2} \color{blue}{(4 x^2 - 2x + 6 x y)} = \color{red}{-\dfrac{1}{2} x^2}\color{blue}{(4 x^2)} \color{red}{-\dfrac{1}{2} x^2}\color{blue}{(-2x)} \color{red}{-\dfrac{1}{2} x^2}\color{blue}{(6xy)}\)
Mulitply constants together and variables together
\( \qquad \qquad = -\dfrac{1}{2} (4) (x^2 x^2) -\dfrac{1}{2} (-2)(x^2 x) -\dfrac{1}{2} (6) (x^2 x y) \)
Simplify
\( \qquad \qquad = - 2x^4 + x^3 - 3x^3 y \)

Multiply Polynomials with Examples

In order to multiply polynomials, we use distribution to write the multiplication as a sum of mutliplication of monomials by polynomials which we have already practiced above.
Example 2
Multiply the following polynomials.
a)   \( (x - 1) (x + 2) \)   b) \( (- 3 x^2 - x) (x^2 - 2x - 1) \)  
c) \( (2 x - y) ( - x - y) \)
Solution to Example 2
a)
Given \( \qquad (x - 1) (x + 2) \)
Use distribution of the form: \( \color{red}{(a + b)} \color{blue}{ c } = \color{red}{a} \color{blue}{c} + \color{red}{b} \color{blue}{c} \) to rewrite the above as
\( \qquad \qquad \color{red}{(x - 1)} \color{blue}{(x + 2)} = \color{red}{x} \color{blue}{(x+2)} \color{red}{-1} \color{blue}{(x+2)} \)
Use the distributive law   \( \color{red}{a} \color{blue}{(b+c)} = \color{red}a \color{blue}b + \color{red}a \color{blue}c \)   to write the right side above as the sum of products of monomials
\( \qquad \qquad = \color{red}{x} \color{blue}{(x)} + \color{red}{x} \color{blue}{(2)} \color{red}{-1} \color{blue}{(x)} \color{red}{-1} \color{blue}{(2)} \)
Multiply
\( \qquad \qquad = x^2 + 2x - x - 2 \)
Group like terms and simplify
\( \qquad \qquad = x^2 + x - 2 \)

b)
Given \( \qquad (- 3 x^2 - x) (x^2 - 2x - 1) \)
Use distribution of the form: \( \color{red}{(a + b)} \color{blue}{ c } = \color{red}{a} \color{blue}{c} + \color{red}{b} \color{blue}{c} \) to rewrite the above as
\( \qquad \qquad \color{red}{(- 3 x^2 - x)} \color{blue}{(x^2 - 2x - 1)} = \color{red}{-3x^2} \color{blue}{(x^2 - 2x - 1)} \color{red}{-x} \color{blue}{(x^2 - 2x - 1)} \)
Use the distributive law   \( \color{red}{a} \color{blue}{(b+c+d)} = \color{red}a \color{blue}b + \color{red}a \color{blue}c + \color{red}a \color{blue}d \)   to write the right side above as the sum of products of monomials
\( \qquad \qquad = \color{red}{- 3 x^2 } \color{blue}{(x^2)} \color{red}{- 3 x^2} \color{blue}{(-2x)} \color{red}{- 3 x^2 } \color{blue}{(-1)} \color{red}{-x} \color{blue}{(x^2)} \color{red}{-x} \color{blue}{(-2x)} \color{red}{-x} \color{blue}{(-1)} \)
Multiply
\( \qquad \qquad = -3x^4 + 6x^3 + 3x^2 - x^3 + 2x^2 + x \)
Group like terms and simplify
\( \qquad \qquad = -3x^4 + 5x^3 + 5x^2 + x \)

c)
Given \( \qquad (2 x - y) ( - x - y) \)
Use distribution of the form: \( \color{red}{(a + b)} \color{blue}{ c } = \color{red}{a} \color{blue}{c} + \color{red}{b} \color{blue}{c} \) to rewrite the above as
\( \qquad \qquad \color{red}{(2 x - y)} \color{blue}{(- x - y)} = \color{red}{2x} \color{blue}{(-x-y)} \color{red}{-y} \color{blue}{(-x-y)} \)
Use the distributive law   \( a(b+c) = ab + ac \)   to write the right side above as the sum of products of monomials
\( \qquad \qquad = \color{red}{2x} \color{blue}{(-x)} + \color{red}{2x}\color{blue}{(-y)} \color{red}{-y} \color{blue}{(-x)} \color{red}{-y} \color{blue}{(-y)} \)
Multiply
\( \qquad \qquad = -2x^2 - 2x y + y x + y^2 \)
Group like terms and simplify (NOTE: \( x y = y x \) )
\( \qquad \qquad = -2x^2 - x y + y^2 \)

Expand Powers of Polynomials with Examples

Example 3
Expand the following and write as polynomials.
a)   \( (x - 1) ^2 \)   b) \( (x + 3)^3 \)
Solution to Example 3
a)
Given \( \qquad \qquad (x - 1) ^2 \)
Write the above as a product of polynomials
\( \qquad \qquad = \color{red}{(x -1)} \color{blue}{(x - 1)} \)
Distribute as \( (a + b) c = a c + b c \)
\( \qquad \qquad = \color{red}x \color{blue}{(x - 1)} \color{red}{- 1} \color{blue}{(x - 1)} \)
Use the distributive law   \( a(b+c) = ab + ac \)   to write the above as the sum of products of monomials
\( \qquad \qquad = x(x) + x (-1) - 1 (x) -1(-1) \)
Multiply
\( \qquad \qquad = x^2 - x - x + 1 \)
Group like terms and simplify
\( \qquad \qquad = x^2 - 2x + 1 \)

b)
Given \( \qquad \qquad (x + 3)^3 \)
Write the above as a product of polynomials
\( \qquad \qquad = (x + 3)\color{red}{(x + 3)} \color{blue}{(x + 3)} \)
Use distribution to multiply the second and third terms
\( \qquad \qquad = (x+3) (\color{red} x \color{blue}{(x + 3)} \color{red} {+3} \color{blue}{(x + 3)}) \)
Use distribution to expand \( \qquad \color{red} x \color{blue}{(x + 3)} \color{red} {+3} \color{blue}{(x + 3)} \)
\( \qquad \qquad = (x+3) (x^2 + 3x + 3x + 9) \)
Group like terms and simplify \( (x^2 + 3x + 3x + 9) \)
\( \qquad \qquad = \color{red}{(x+3)} (x^2 + 6x + 9) \)
Distribute \( (x^2 + 6x + 9) \)
\( \qquad \qquad = \color{red}x(x^2 + 6x + 9)+\color{red}3(x^2 + 6x + 9) \)
Distribute \( x \) and \( 3 \)
\( \qquad \qquad = x^3 + 6x^2 + 9x + 3x^2 + 18x + 27 \)
Group like terms and simplify
\( \qquad \qquad = x^3 + 9x^2 + 27x + 27 \)

Questions

1. Multiply the following polynomials.
   
   
   1. ) \( ( - 2) (- 2 x) \)
      
   2. ) \( (x) (x + 1) \)
      
   3. ) \( - x^2 (- x + 1) \)
      
   4. ) \( (- 4 x^3 - x) (2x - 1) \)
      
   5. ) \( (- 4 x^3 - y) (2x - y) \)
      
   6. ) \( (- 7 x^2 - 2x + 3) (2x^2 - x + 2) \)
      
   7. ) \( (- \dfrac{1}{3} x^2 + 4) (- \dfrac{1}{2} x + 9) \)
   
   
2. Use multiplication and addition of polynomials to write as a single polynomial.
   
   
   1. ) \( (2x-1)(3x-2) + 3x - 9 \)
      
   2. ) \( -(2x + 2) - (2x - 1)(x - 3) \)
      
   3. ) \( (x^2 - 1)(x - 2) - (x - 3)(2x^2 - 4)) \)
      
   4. ) \( (-3x - 2)(y - 3) + (x - 5)(y - 6) \)
      
   
   
3. Use multiplication of polynomials to expand the following.
   
   
   1. ) \( (x + 3 y)^2 \)
      
   2. ) \( (2 x - y)^2 \)
      
   3. ) \( (x - y)(x +y) \)
      
   4. ) \( (x - 3)^3 \)

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