Factorise completely: 32x2y - 48x3y3

Question: Factorise completely: 32x²y − 48x³y³

(The question is read as 32x²y − 48x³y³, since the "2" and "3" are exponents.)

Step 1: Analyse the question "Factorise completely" means we must take out the Highest Common Factor (HCF) of both terms, so nothing further can be taken out of the bracket.

Step 2: Find the HCF

• Numbers: HCF of 32 and 48 is 16 (32 = 16 × 2, 48 = 16 × 3).
• x: the lowest power in x² and x³ is x².
• y: the lowest power in y and y³ is y.

HCF = 16x²y

Step 3: Divide each term by the HCF

• 32x²y ÷ 16x²y = 2
• 48x³y³ ÷ 16x²y = 3xy²

Step 4: Write the answer

32x²y − 48x³y³ = 16x²y(2 − 3xy²)

Step 5: Check by expanding 16x²y × 2 = 32x²y ✓ 16x²y × 3xy² = 48x³y³ ✓

Why the other options are wrong

• B: 8xy(4x − 6x²y²) is correct when expanded, but it is not fully factorised, because 2x can still be taken out of the bracket.
• C: 8x²y(4 − 6xy²) is also not complete, because 2 can still be taken out of the bracket.
• D: 16xy(2x − 3x²y²) still has a common factor of x inside the bracket.

Tip: After factorising, always check the bracket. If its terms still share a common factor, you haven't factorised completely. Take the lowest power of each letter and the largest number that divides both coefficients.

The correct answer is A.