Multiplying Two Binomials with an Area Model

0:00How do we multiply x plus 3 by x plus 5?

0:05Each bracket is a binomial: an expression with two terms, like x and 3.

0:13Draw a rectangle that is x plus 5 wide and x plus 3 tall. Its area is the product.

0:21Split the width into x and 5, and the height into x and 3.

0:26x times x is x squared.

0:30x times 5 is 5x.

0:333 times x is 3x.

0:36And 3 times 5 is 15.

0:39The whole area is the sum of the four parts: x squared plus 5x plus 3x plus 15.

0:485x and 3x are like terms: both have x to the same power. Together they make 8x.

0:56So x plus 3, times x plus 5, equals x squared plus 8x plus 15.

1:04Check with x equals 2: 5 times 7 is 35, and 4 plus 16 plus 15 is also 35.

1:13This is the distributive property: multiply the outside term by each term inside. So x times x plus 5 is x squared plus 5x.

1:25And 3 times x plus 5 is 3x plus 15.

1:30A common mistake is to multiply only the first terms and the last terms: x squared plus 15. It misses 5x and 3x.

1:41Now a negative term: x minus 2, times x plus 6. Keep the minus sign with the 2.

1:49The four parts are x squared, 6x, minus 2x and minus 12, as a negative times a positive is negative.

1:596x minus 2x is 4x, so the product is x squared plus 4x minus 12.

2:07To multiply two binomials, multiply each term by each term, then add the like terms.