Multiplication of polynomials

Multiplication of polynomials:

there are two formats for this: horizontal and vertical, like in addition.

The simplest case of multiplication of polynomials is multiplication of monomials.

For instance :

Simplify: ( 5 x² )(-2x ³)

For multiplying these two monomials  we have to just multiply the numbers and add the powers using the exponent rule.

So (-6x²)(3x³) = -18 x² ⁺ ³= -18 x⁵

Our next one is multiplying a monomial with a polynomial

For example: -5 x (x ⁴ -5 x³ + 2 x - 9)

Here, we have to multiply the monomial, with each term of the polynomials.

That can be done as in the previous example.

           = (-5 x) (x⁴ )+(-5 x) (-5x³)+(-5 x) (2 x) + (-5 x) (-9)

           = -5x⁴⁺¹ +25 x¹⁺ ³ - 10x¹ ⁺ ¹ + 45 x

           = -5 x ⁵ +25 x⁴ - 10 x ² + 45 x

The next one is two terms times two terms; that is multiplication of binomial by another binomial.

There are three methods to do this. We will see one by one.

Example using horizontal method in multiplication of polynomials

The first method is horizontal method

For example:

(2x+3) (x-5)

Here, we have to distribute twice taking each term in the first bracket,”through” each term in the second parentheses.

(2x+3) (x-5)

          = (2x+3)x + (2x+3)(-5)

          = (2x) x + (3) x + (2x) (-5) + (3) (-5)

          = 2 x ² + 3 x - 10 x - 15

         = 2x ² -7 x - 15

The horizontal method is the little difficult one, so let us see the second method ,the vertical method which is more simpler.

Like how we do number using vertical method, (we are stacking the numbers vertically, start from the right to left)

Took one term at a time, and start multiply the terms  from the right to left, and we have to arrange them in a row, and move to the next term and arrange the multiplied term in the next row. Abusing the rows to the left as we work from term to term, in the lower polynomial

Then we add them column wise, and get the final polynomial.

For example, simplify (2x + 3) (x - 5)

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