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Let's look at our objectives for today. We'll start by looking at how we use the acronym FOIL. We'll then look at how we factor quadratics. And finally, we'll do some examples factoring quadratics.
Let's start by reviewing how to use FOIL. FOIL is used to multiply two binomials together. FOIL is an acronym that we use to remember the terms to multiply. It stands for First, Outside, Inside, and Last.
For example, if we want to multiply the binomials x plus 3 times x plus 5, we start by multiplying our first two terms in each parenthesese. X times x gives us x squared.
We then multiply our outside terms x times 5 is 5x, our inside terms, 3 times x, gives us 3x, and our last two terms, 3 times 5, gives us 15. We then combine our two like terms, 5x plus 3x is 8x. So our final answer is x squared plus 8x plus 15. The resulting expression is a quadratic expression and can be generally written in this form, ax squared plus bx plus c.
Quadratic expressions in this form have an x squared term, an x term, and a constant term. Quadratic factoring is the reverse process of FOILing. Therefore, we could factor our expression x squared plus 8x plus 15, and the result would be our original expression x plus 3 times x plus 5. Factoring quadratic expressions is useful when solving quadratic equations and provides important information about the graphs of quadratic equations.
Let's use our previous example to look at the process for factoring quadratics. We can see that the constant term 15 in expanded form is the product of the two numbers in factored form, 3 and 5. We can also see that the coefficient of the x term in expanded form, , 8, is the sum of the two numbers in factored form. When we factor a quadratic expression in expanded form we use these two patterns to identify the two numbers to use in the factored form of the expression.
So let's see what that looks like. To write our expanded form expression into factored form, we start by identifying the pairs of numbers that, when multiplied together, equal the constant term 15. When considering factor pairs for a constant term, both positive and negative numbers should be considered. This will especially be important when there is a subtraction in the expanded form, as this indicates that at least one number in the factor pair will be negative.
So the factor pairs of 15 are 1 and 15, negative 1 and negative 15, 3 and 5, and negative 3 and negative 5. We then add our factor pairs to find the pair of numbers that also add to the coefficient of our x term 8. 1 and 15 have the sum of 16. Negative 1 and negative 15 have a sum of negative 16. But 3 and 5 have a sum of 8.
Once we have found the pair of numbers with the correct sum, we do not need to consider other factor pairs, because there will be only one pair of numbers that will multiply to the constant term and add to the x term coefficient. So 3 and 5 are our pair of numbers that multiply to 15 and add to 8. We finally use this pair of numbers to write the expression in factored form. So we have x plus 3 times x plus 5.
Let's do some examples. We want to factor the expression x squared plus 4x minus 12. We start by identifying the pair of numbers that when multiplied together equal negative 12. Because our constant term is negative, we know that one of the numbers must be negative and the other one will be positive. The pairs of numbers that multiply to negative 12 are negative 1 and 12, 1 and negative 12, negative 2 and 6, 2 and negative 6, negative 3 and 4, and 3 and negative 4.
We next look for which pair of these numbers also have a sum of 4. Negative 1 and 12 add to 11. 1 and negative 12 add to negative 11. Negative 2 and 6 add to 4. So now we can use our two numbers, negative 2 and 6, to write our expression in factored form as x minus 2 times x plus 6.
We can verify that we correctly factored expression by multiplying the binomials together using FOIL to see that we arrive back at our original expression. So we multiply our two binomials together using FOIL. X times x gives us x squared. X times 6 gives us 6x. Negative 2 times x gives us negative 2x. And negative 2 times 6 gives us negative 12.
We combine our like terms negative 2x and 6x, which gives us x squared plus 4x minus 12. This is the same as our original expression, which means that x minus 2 times x plus 6 is the correct factored form.
Let's do another example. We want to factor the expression x squared minus 7x plus 10. We start by identifying the pairs of numbers that, when multiplied together, equal 10. Because the pair of numbers must multiply to a positive number but add to a negative number, both of the numbers must be negative. The pairs of numbers that multiply to 10 are negative 1 and negative 10 and negative 2 and negative 5.
We now look for which of these pairs of numbers also have a sum of negative 7. Negative 1 and negative 10 add to negative 11. But negative 2 and negative 5 add to negative 7. So we use negative 2 and negative 5 to write our expression in factored form as x minus 2 times x minus 5.
Let's do one last example. We want to factor the expression x squared minus x minus 2. We start by identifying the pairs of numbers that, when multiplied together, equal negative 2. Because the pair of numbers must multiply to a negative number and add to a negative number, one of the numbers will be positive and one will be negative.
The pairs of numbers that multiply to negative 2 are negative 1 and 2 and 1 and negative 2. Because there's no written number in front of the x term, the x term has an implied coefficient of negative 1. So we look for which of these pairs of numbers also have the sum of negative 1. Negative 1 and 2 add to positive 1. But 1 and negative 2 add to a negative 1. So we use 1 and negative 2 to write our expression in factored form as x plus 1 times x minus 2.
Let's review our important points from today. Make sure you get these in your notes so you can refer to them later. Quadratic factoring is used to write a quadratic expression from expanded form to factored form. The constant term in expanded form is the product of the two numbers in factored form. And the coefficient of the x term in expanded form is the sum of the two numbers in factored form. We use these two facts to write our expression from expanded form to factored form.
So I hope that these key points and examples helped you understand a little bit more about basic quadratic factoring. Keep using your notes, and keep on practicing, and soon you'll be a pro. Thanks for watching.