SAT-Math-29: Factoring: Greatest Common Factor and Grouping

1. Extracting the Greatest Common Factor (GCF)

Factoring is simply the reverse of multiplication. The very first step when factoring any algebraic expression on the SAT is to look for the Greatest Common Factor (GCF). This is the largest number and the highest power of variables that divides evenly into every single term.

Once identified, you pull the GCF outside a single set of parentheses, dividing each original term by that factor.

• Example: Factor $6x^3 + 12x^2$.

  • The largest integer that divides 6 and 12 is 6.

  • The highest power of $x$ shared by both terms is $x^2$.

  • Therefore, the GCF is $6x^2$.

  • Pull it out: $6x^3 + 12x^2 = 6x^2(x + 2)$.

2. Factoring by Grouping

When you face a polynomial with four terms, finding a single GCF for the whole expression is often impossible. Instead, we use Factoring by Grouping (Guruhlash usuli).

The Grouping Blueprint

1. Split the four-term polynomial into two separate pairs.

2. Factor out the unique GCF from the first pair, and then from the second pair.

3. If done correctly, the remaining expressions inside the parentheses will match perfectly.

4. Factor out that shared parentheses chunk as a single binomial factor.

🇺🇿 Uzbek Explanation:

Guruhlash usuli (Grouping): Agar ko'phadda 4 ta had qatnashgan bo'lsa, ularni ikkita alohida guruhga (juftlikka) ajratamiz.

Har bir juftlikdan umumiy ko'paytuvchini qavsdan tashqariga chiqaramiz. Agar to'g'ri bajargan bo'lsangiz, qavs ichidagi ifodalar mutlaqo bir xil bo'lib qoladi. Keyin o'sha bir xil qavsni bitta guruh qilib oldinga chiqaramiz.

3. High-Yield SAT Practice Question

Question: Which of the following is a completely factored form of the expression $x^3 + 3x^2 + 5x + 15$?

A) $(x^2 + 3)(x + 5)$

B) $(x^2 + 5)(x + 3)$

C) $x^2(x + 3) + 5(x + 3)$

D) $(x + 3)^2(x + 5)$

Step-by-Step Explanation:

1. Split the polynomial into two pairs:

   $$(x^3 + 3x^2) + (5x + 15)$$

2. Extract the GCF from each pair separately:

   • For the first pair $(x^3 + 3x^2)$, the GCF is $x^2 \rightarrow x^2(x + 3)$

   • For the second pair $(5x + 15)$, the GCF is $5 \rightarrow 5(x + 3)$

3. Rewrite the expression:

   $$x^2(x + 3) + 5(x + 3)$$

   (Notice how the $(x + 3)$ matches perfectly!)

4. Factor out the shared $(x+3)$ block: Pull $(x + 3)$ to the front, leaving the remaining outside coefficients inside their own bracket:

   $$(x^2 + 5)(x + 3)$$

Correct Answer: B

Summary

• GCF priority: Always scan an expression for a universal common factor before trying any other factoring technique.

• Four-term trigger: When you spot exactly 4 terms, your automatic reflex should be factoring by grouping.

• The structural check: Grouping is only successful if the binomial inside the brackets matches perfectly across both halves.

Superb work, my bro! Let's keep moving through this factoring suite. Ready for SAT-Math-30: Factoring: Difference of Squares and Perfect Square Trinomials?