**Multiplying polynomials by a monomial :**

We have two methods to find the product of polynomial by a monomial.

i) Distributive

ii) Box method

In the first method, we need to distribute the monomial with each and every terms of the polynomial.

In the second method, we need to draw a table and fill up the terms and multiply.

If we want to multiply a monomial by a monomial, we have to follow the order given below

(i) Multiply the signs

(ii) Multiply the variables

(iii) Multiply the numbers

Let us discuss the concept in the below examples.

**Example 1 :**

Multiply 2 x^{3} (3x + 2y)

**Solution :**

**Distribution method :**

**Box method :**

Here you can find the another method of multiplying polynomial by a monomial. In both ways, we will get the same answer.

**Example 2 :**

Multiply 5 x^{2} (3x + 2y - 5z)

**Solution :**

= 5 x^{2} (3x + 2y - 5z)

= 5 x² (3x) + 5 x²(2y) + 5 x²(-5z)

= 5 (3) x² x + 5(2) x²y + 5(-5) x²z

= 15 x^{(2+1)} x + 10 x²y - 25 x²z

= 15 x³ + 10 x²y - 25 x²z

**Box method :**

Here we have done the same problem using box method. In both ways, we will get the same answer.

**Example 3 :**

Multiply (-7 u) (u + 3 u²)

**Solution :**

**Distribution method :**

= (-7 u) (u + 3 u²)

= (-7 u) (u) + (-7 u)(3 u²)

= -7 u^{(1+1)} - 21 u^{(1+2)}

= -7 u² - 21 u³

**Box method :**

Here we have done the same problem using box method. In both ways, we will get the same answer.

**Example 4 :**

Multiply (2x) (3x² + 5x + 7)

**Solution :**

= (2x) (3x² + 5x + 7)

= (2x) (3x^{2}) + (2x)(5x) + (2x)(7)

= 6x^{3} + 10x^{2} + 14x

**Example 5 :**

Multiply (4m²n) (7n - 3m-5mn)

**Solution :**

= (4m²n) (7n - 3m - 5mn)

= (4m²n) (7n) + (4m²n)(- 3m) + (4m²n)(-5mn)

= 28 m²n² -12 m³n - 20 m³n²

**Example 6 :**

Multiply 3a (4a-7b)

**Solution :**

= 3a (4a - 7b)

= 3a (4a) + 3a (-7b)

= 12a²- 21 a b

**Example 7 :**

Multiply the polynomial (3x³+4x²−5) by the monomial (7x)

**Solution :**

= 7x (3x³+ 4x²−5)

= 7x (3x³) + 7x (4x²) + 7x (-5)

= 21 x^{(1+3)} + 28 x^{(1+2)} - 35x

= 21 x^{4} + 28 x^{3} - 35x

**Example 8 :**

Multiply the polynomial (3x+ 7x²) by the monomial (3x)

**Solution :**

= 3x (3x+ 7x²)

= 3x (3x) + 3x (7x²)

= 9x^{2} + 21 x^{3}

**Example 9 :**

Multiply the polynomial (5x + 7xy + 3x^{2}) by the monomial (6x)

**Solution :**

= 6x (5x + 7xy + 3x^{2})

= 6x (5x) + 6x (7xy) + 6x(3x^{2})

= 30x + 42x^{2}y + 18x^{3}

- Multiplying polynomials using FOIL method
- Multiplying polynomial using BOX method
- Multiply and divide monomials
- Multiplying polynomials examples
- Adding and subtracting polynomials
- Factoring polynomials
- Dividing polynomials using long division
- Add and subtract polynomials using algebra tiles

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