# Multiplying polynomials and binomials

We can use the area of a rectangle to explain how you multiply a polynomial by a monomial.

Example:

Find the area of this rectangle.

$\\A=b\cdot h \\\\A=5x\left ( 5x-4 \right ) \\A=5x\cdot 5x-5x\cdot 4 \\A=5\cdot 5\cdot x\cdot x-5\cdot 4\cdot x \\\\A=25x^{2}-20x$

This method is called the distributive property. The distributive property shows us how to write an expression in a different way.

$\\a(b+c)=ab+ac\\$

Example:

With numbers

$\\5\left (2+6 \right )=5\cdot 2+5\cdot 6=10+30=40\\$

With variables and numbers:

$\\7x+4x= x\left (7+4 \right )=11x \\$

We can use the area of another rectangle to explain what happens when you multiply two binomials.

Example:

The area of the rectangle can be calculated by the use of the distributive property:

$\\A=b\cdot h\\\\ A=({\color{red} 4x}+{\color{blue} 3})({\color{green} 2x}+2)\\\\ A=({\color{red} 4x}+{\color{blue} 3})\cdot {\color{green} 2x}+({\color{red} 4x}+{\color{blue} 3})\cdot 2\\\\ A={\color{red} 4x}\cdot {\color{green} 2x}+{\color{blue} 3}\cdot {\color{green} 2x}+{\color{red} 4x}\cdot 2+{\color{blue} 3}\cdot 2\\\\ A=8x^{2}+6x+8x+6\\\\ A=8x^{2}+14x+6\\$

Video lesson: Expand the expression

$(3x+4)(x^{2}-2)$