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Multiplying linear expressions
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What does the graph of the product of two linear expressions look like?
When you draw a straight line, you write it like this: The slope multiplied by a variable, ... and a constant term. When expressions have one variable... ... and the variable does not have a number up here - as with these lines we've drawn - then the expressions are linear. Now, let's multiply two of these linear algebraic expressions with each other.
Let's take 'x' plus 'three' multiplied by 'x' plus 'two'. The important thing here is that all the terms in the first bracket should be multiplied by all the terms in the second. It's important to be thorough. Let's begin with the first term in the first bracket. We multiply it by the first term in the second bracket, and then by the second term in the second bracket.
And take the second term in the first bracket, and multiply that with the first term in the second bracket and then with the second. And now all the terms have been multiplied by each other. Time to clean up the expression. Pause, and simplify it, for yourself. Right. 'X squared' plus 'five x' plus 'six'.
Let's try another one. 'Five' minus 'x' multiplied by 'two x' plus 'three'. There are two things to remember here: The subtraction sign, and that the x comes as second term. Let's solve this just like before. Thoroughly. Pause the film and try it yourself. '5' multiplied by 'two x' ...
is 'ten x'. Plus '5' multiplied by '3'... ... 15. Remember the subtraction sign. 'Minus multiplied by plus' is 'minus', so 'minus x' multiplied by 'two x' is minus 'two x-squared'. And then 'minus x' multiplied by '3' ...
is equal to minus 'three x'. Time to simplify again. Combine the terms that are the “same kind”... and what's left is: Minus 'two x-squared' plus 'seven x' plus '15'. There, you see?
It works, as long as you're thorough. You probably noticed: An 'x squared' term comes up every time we multiply two linear expressions. Think of the linear expression as a straight line... that we'd measure in meters. Linear algebraic expression.
When we add another line, it becomes a line multiplied by a line. An area... that we'd measure in square meters. What we have here, is a squared algebraic expression. In addition, they look completely different as graphs.
As soon as we have a square expression we get a curve on the graph. - a quadratic equation - Let's take the last one! Now, there are two subtraction signs, so you need to be extra careful. 'Three' minus 'x' multiplied by 'four' minus 'x' As usual, pause, and try it yourself. ... Now, how did it work with 'minus' multiplied by 'minus' ? Exactly... it becomes 'plus'.
Plus 'x-squared' And simplify... It becomes: 'x-squared' minus 'seven x' plus '12' Now we've multiplied various linear algebraic expressions.There's a pattern in them. There's a pattern in them. Mathematics is often about patterns! Let's see if we can find the pattern here, shall we?
Look! Do you see it? This one is always this one plus this one' and this one is always this one multiplied by that. And be careful with the subtraction signs! Watching for this pattern is good for self-checking during a test...
But you will also make use of it later. When you're factoring and solving quadratic equations. Be thorough and make sure that all terms are multiplied with all terms. Simplify the expression and make it neat. Remember the pattern: In front of the x term, these two are added to each other The constant term is these two multiplied by each other.
And be careful with the signs. Practice on your own, and it'll get easier to find the pattern. And notice that it always becomes a quadratic equation when you multiply two linear expressions with each other.