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

In the first part, we discuss how we multiply a monomial by a polynomial using the distributive law and then extend the same idea to the multiplication of polynomials.

## Multiply Monomials by Polynomials with Examples

To
multiply polynomials , you need to know how
1) to use the distributive law: $\quad a(b+c) = ab + ac \quad$ or $\quad (b+c) a = b a + c a \quad$, which is one of the
basic rules of algebra ,

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

Solutions are also included.

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. ) $(x - 3)^3$

The questions and their solutions to the above questions are included.