How To Factor The Greatest Common Factor In a Polynomial | Algebra

Summary

This video tutorial explains how to factor out the greatest common factor (GCF) in various algebraic expressions. It starts with simple examples, such as factoring out the GCF of 12x and -20, progressing to more complex terms like polynomials and binomials. The video demonstrates step-by-step techniques for identifying the GCF and simplifying expressions by dividing both terms by it. It covers cases with different variables and coefficients, showing how to factor out common terms, combine like terms, and even further simplify when possible. Viewers will gain a solid understanding of factoring techniques to solve algebraic expressions effectively.

Takeaways

• 😀 The Greatest Common Factor (GCF) is the largest number that divides each term in a given expression.
• 😀 When factoring out the GCF from a binomial, start by identifying the highest common factor of the coefficients and variables.
• 😀 The GCF of 12x and -20 is 4, which simplifies the expression to 4(3x - 5).
• 😀 For 6x² and 18x, the GCF is 6x, resulting in 6x(x + 3).
• 😀 For 14x² and -35x, the GCF is 7x, simplifying the expression to 7x(2x - 5).
• 😀 The GCF for 36x⁵ and -63x³ is 9x³, giving the result 9x³(4x² - 7).
• 😀 To factor out a GCF from a trinomial (e.g., 2x² - 8x + 10), first identify the common factor of the coefficients, which is 2.
• 😀 The GCF for 27x⁵, 36x³, and -45x² is 9x², which simplifies to 9x²(3x³ + 4x - 5).
• 😀 When factoring expressions like x(x - 2) + 3(x - 2), the common factor (GCF) is (x - 2), simplifying to (x - 2)(x + 3).
• 😀 In expressions with two variables (e.g., y(3x - 2) - 4(3x - 2)), the GCF is the common factor (3x - 2), resulting in (3x - 2)(y - 4).

Q & A

What is the first step in factoring out the greatest common factor (GCF) in an expression?

The first step is to identify the greatest number that divides both coefficients of the terms in the expression. Once the number is found, you also need to check for any common variables between the terms.

In the example 12x - 20, how is the GCF determined?

The GCF of 12 and 20 is 4, as it is the highest number that divides both of them evenly. Therefore, the expression can be factored as 4(3x - 5).

Why is 4 the GCF of 12 and 20 in the example 12x - 20?

Both 12 and 20 are divisible by 2 and 4, but 4 is the greatest of these factors, making it the GCF.

What do we do after factoring out the GCF from an expression like 6x² + 18x?

After factoring out the GCF (in this case, 6x), you divide each term by the GCF to simplify the expression, resulting in 6x(x + 3).

How do you handle an expression like 36x⁵ - 63x³ when factoring out the GCF?

You first find the GCF, which is 9x³ in this case, and then divide each term by this factor. The factored form is 9x³(4x² - 7).

In the expression 2x² - 8x + 10, what is the GCF, and how do you factor it out?

The GCF of 2x², -8x, and 10 is 2. Factoring out the 2, you get 2(x² - 4x + 5), but the trinomial inside cannot be factored further.

What do you do when a trinomial like x² - 4x + 5 cannot be factored?

If you cannot find two numbers that multiply to 5 and add to -4, you leave the trinomial as is and do not factor it further.

How is the GCF factored out from the expression 27x⁵ + 36x³ - 45x²?

The GCF of 27, 36, and 45 is 9, and the GCF of the variable terms is x². The expression becomes 9x²(3x³ + 4x - 5).

What happens in the example x(x - 2) + 3(x - 2)?

The GCF here is (x - 2), and factoring it out results in (x - 2)(x + 3).

In the expression 2x - 3)(y - 4) + (3x + 8)(y - 4), how is the GCF identified, and how do we simplify the expression?

The GCF is (y - 4). Factoring it out, you get (y - 4)(2x - 3 + 3x + 8). After combining like terms, the expression simplifies to 5x + 5, and factoring out the 5 gives the final answer as 5(x + 1)(y - 4).