# 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.

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 questionsand their solutions to the above questions are included.

## More References and Links

Solutions to Mulitply Polynomials.
Multiply and Simplify Monomials
Add and subtract Polynomials
Exponents Questions
Middle School Math (Grades 6, 7, 8, 9) - Free Questions and Problems With Answers
High School Math (Grades 10, 11 and 12) - Free Questions and Problems With Answers
Primary Math (Grades 4 and 5) with Free Questions and Problems With Answers