To factor in mathematics is to express a quantity as the product of two or more quantities. Take the integer 8 for example. Two factors of 8 are 2 and 4. We can express this as an equation: 2 x 4 = 8. There are two types of expressions you will need to factor on the SAT: polynomials and quadratics.
A polynomial is a numerical expression with more than one term. For example, 4x + 2x. If we were asked to factor this expression, we would want to see what is common to both terms and then divide it from each term. Right away, we see that both terms contain x. We also notice that one of the terms contains 2 and that 2 is a factor of the coefficient of the first term, 4. So we know we can factor out 2x from this equation. Once we’ve found one of the factors, we put it in parentheses and leave what is left behind in another set of parentheses. Here’s how it would look:
4x + 2x = (2x)(2 + 1)
We must have a 1 left in the second parentheses to act as a place-holder for the 2x. A quick way to check your work is to distribute the 2x across the terms and make sure you get 4x + 2x. It’s important to make sure you have the correct factors and this extra step is a quick way to make sure you don’t make any simple mistakes.
The quadratic equation is a special polynomial. It has three terms and is in the form ax² + bx + c. The first term will always be squared, the middle term will always have the same variable, and the last term will contain only an integer.
An example of a quadratic is x² – 5x + 6. To find the factors of this equation, we must set up our set of two parentheses:
( )( )
The first term in both parentheses must be x, since x multiplied by x is the only way to get x². Then we look at the coefficient of the second term, -5. It’s important to include the sign in front of the integer as part of the coefficient. One of the rules of quadratic equations is that the second terms in the two factors must add together to equal the middle term’s coefficient. So we need to think of two numbers that add together to give us -5.
Already, we can think of many combinations: -6 and 1, -2 and -3, -200 and 105. So which pair is it? Now we have to look at the integer that’s the third term of the quadratic. Here it’s + 6. Another rule of quadratic equations is that the third term of the quadratic equation will equal the product of the second terms in the two factors. So not only do we need the two numbers to add together to equal -5, but we need them to multiply together to equal + 6. Therefore the factors must be:
(x – 2) (x – 3)
Those are the two “factors” of our original equation: x² – 5x + 6. If an SAT question asks for the roots or solutions of a quadratic equation, you would simply set the two factors each equal to zero and solve for x.
(x – 2) = 0 (x – 3) = 0
+2 +2 +3 +3
x = 2 x = 3
Therefore the “roots” or “solutions” to this quadratic are 2 and 3. For more practice with factoring, create your own game on Grockit and choose to focus on Algebra questions!
To factor in mathematics is to express a quantity as the product of two or more quantities. Take the integer 8 for example. Two factors of 8 are 2 and 4. We can express this as an equation: 2 x 4 = 8. There are two types of expressions you will need to factor on the SAT: polynomials and quadratics.
A polynomial is a numerical expression with more than one term. For example, 4x + 2x. If we were asked to factor this expression, we would want to see what is common to both terms and then divide it from each term. Right away, we see that both terms contain x. We also notice that one of the terms contains 2 and that 2 is a factor of the coefficient of the first term, 4. So we know we can factor out 2x from this equation. Once we’ve found one of the factors, we put it in parentheses and leave what is left behind in another set of parentheses. Here’s how it would look:
4x + 2x = (2x)(2 + 1)
We must have a 1 left in the second parentheses to act as a place-holder for the 2x. A quick way to check your work is to distribute the 2x across the terms and make sure you get 4x + 2x. It’s important to make sure you have the correct factors and this extra step is a quick way to make sure you don’t make any simple mistakes.
The quadratic equation is a special polynomial. It has three terms and is in the form ax² + bx + c. The first term will always be squared, the middle term will always have the same variable, and the last term will contain only an integer.
An example of a quadratic is x² – 5x + 6. To find the factors of this equation, we must set up our set of two parentheses:
( )( )
The first term in both parentheses must be x, since x multiplied by x is the only way to get x². Then we look at the coefficient of the second term, -5. It’s important to include the sign in front of the integer as part of the coefficient. One of the rules of quadratic equations is that the second terms in the two factors must add together to equal the middle term’s coefficient. So we need to think of two numbers that add together to give us -5.
Already, we can think of many combinations: -6 and 1, -2 and -3, -200 and 105. So which pair is it? Now we have to look at the integer that’s the third term of the quadratic. Here it’s + 6. Another rule of quadratic equations is that the third term of the quadratic equation will equal the product of the second terms in the two factors. So not only do we need the two numbers to add together to equal -5, but we need them to multiply together to equal + 6. Therefore the factors must be:
(x – 2) (x – 3)
Those are the two “factors” of our original equation: x² – 5x + 6. If an SAT question asks for the roots or solutions of a quadratic equation, you would simply set the two factors each equal to zero and solve for x.
(x – 2) = 0 (x – 3) = 0
+2 +2 +3 +3
x = 2 x = 3
Therefore the “roots” or “solutions” to this quadratic are 2 and 3. For more practice with factoring, create your own game on Grockit and choose to focus on Algebra questions!
