Math agc1 how to do problems and definitions

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Factoring: #1 Identify the greatest common factor:
Identify the greatest common factor in the expression. Factoring is a way to simplify expressions by removing factors that are common across all the terms in the expression. To start, find the greatest common factor that all of the terms in the expression share - in other words, the largest number by which all the terms in the expression are evenly divisible.

Example: 9x to the power of 2 + 27x - 3. Every term in this equation is divisible by 3. Since the terms aren't all evenly divisible by any larger number, we can say that 3 is our expression's greatest common factor.

Factoring #2: Dividing the terms by the GCF:
Divide every term in your equation by the greatest common factor. The resulting terms will all have smaller coefficients than in the original expression.

Factor out 9x ² + 27x -3 by its greatest common factor of 3. To do so divide each term by 3.
1. 9x² divided by 3 = 3x²
Equals: 27x divided by 3 = 9x
Equals: -3 divided by 3 = -1
Thus, our new expression is 3x2 + 9x - 1.

Factoring #3: Make new expression equal to the old one:
To make our new expression equal to the old one, we'll need to account for the fact that it has been divided by the greatest common factor. Enclose your new expression in parentheses and set the greatest common factor of the original equation as a coefficient for the expression in parentheses.

For the example expression, 3x² + 9x - 1:
Enclose the expression in parentheses and multiply by the greatest common factor of the original equation to get 3(3x (² + 9x - 1). This equation is equal to the original, 9x² + 27x - 3.
an algebraic expression that includes a root. The root may be a square root, a cube root, or any other power. Simplifying a radical expression can help you solve an equation. Simplifying radical expressions involves removing the root when possible, or reducing the radicands, the numbers inside the radical symbol, as much as you can.
Simplify any radical expressions that are perfect squares:
A perfect square is the product of any number that is multiplied by itself, such as 81, which is the product of 9 x 9. To simplify a radical expression that is a perfect square, simply remove the radical sign and write the number that is the square root of the perfect square.
For example, 121 is a perfect square because 11× 11 is 121. You can simply remove the radical sign and write 11 as your answer.

Simplify any radical expressions that are perfect cubes:
A perfect cube is the product of any number that is multiplied by itself twice, such as 27, which is the product of 3× 3 × 3. To simplify a radical expression when a perfect cube is under the cube root sign, simply remove the radical sign and write the number that is the cube root of the perfect cube.
For example, 512 is a perfect cube because it is the product of 8 × 8 × 8. Therefore, the cube root of the perfect cube 512 is simply 8.

Break down an imperfect radical expression into its multiples:
STEP 1: The multiples are the numbers that multiply to create a number -- for example, 5 and 4 are two multiples of the number 20. To break down an imperfect radical expression by its multiples, write down all of the multiples of that number (or as many as you can think of, if it's a large number) until you find one that is a perfect square.
For example, all the multiples of the number 45: 1, 3, 5, 9, 15, and 45. 9 is a multiple of 45 that is also a perfect square. 9 × 5 = 45.

STEP 2: Remove any multiples that are a perfect square out of the radical sign: 9 is a perfect square because it is the product of 3 × 3. Take the 9 out of the radical sign and place a 3 in front of it, leaving 5 under the radical sign.

Find a perfect square in the variable
The square root of a to the second power would be a. The square root of a to the third power is broken down into the square root of a squared times a -- this is because you add exponents when you multiply variables, so that a squared times a reverts back to a cubed. Therefore, the perfect square in the expression a cubed is a squared.

Simplify a radical expression with variables and numerals that is a perfect square:
EXAMPLE: √36a²
take apart the expression by first looking for perfect squares in the numbers, and then looking for perfect squares in the variables. Then, remove the radical sign and let their square roots remain. the square root of 36 x a squared.

1. 36 is a perfect square because 6 × 6 = 36.
2. a squared is a perfect square because a times a is a squared.

Now that you've broken down these numbers and variables into their square roots, simply remove the radical sign and leave the square roots. The square root of 36 ×∧ a 6a.

Simplify a radical expression with variables and numerals that is not a perfect square:
√ 50a³

break down the expression into numerals and variables, and search for perfect squares within the multiples of both. Then, pull any perfect squares out of the radical expression.

-Break down 50 to find any multiples that are a perfect square. 25 × 2 = 50 and 25 is a perfect square because 5 × 5 = 25. To simplify root 50, you can pull a 5 out of the radical sign and leave 2 within it.

-Break down "a" to the third power to find any multiples that are a perfect square. a to the third power is really a squared times a, and a squared is a perfect square. You can pull one a out of the radical sign and leave one a inside the sign. Therefore, root a cubed is really a root a.

-Put it all together. Just place everything you took out of the radical sign out of the radical sign and keep everything that you kept in the sign in the sign. Combine 5 root 2 and a root a to create 5 x a root 2 x a
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