Question

factor out a greatest common factor
① $36x^4y^3 - 54x^2y^2 + 90xy$
② $15x^{12} - 35x^4 + 20x^2 + x$
③ $39x^3 + 21x$
④ $54x^{12} - 40x^2$
⑤ $80x^6y^2 - 24xy$

Explanation:

Problem ①: \( 36x^4y^3 - 54x^2y^2 + 90xy \)

Step1: Find GCF of coefficients

Factors of \( 36 \): \( 1, 2, 3, 4, 6, 9, 12, 18, 36 \)
Factors of \( 54 \): \( 1, 2, 3, 6, 9, 18, 27, 54 \)
Factors of \( 90 \): \( 1, 2, 3, 5, 6, 9, 10, 18, 30, 45, 90 \)
GCF of \( 36, 54, 90 \) is \( 18 \).

Step2: Find GCF of variables

For \( x \): lowest power is \( x^1 \) (from \( 90xy \)).
For \( y \): lowest power is \( y^1 \) (from \( 90xy \)).
GCF of variables: \( xy \).

Step3: Factor out GCF

\( 36x^4y^3 - 54x^2y^2 + 90xy = 18xy(2x^3y^2 - 3xy + 5) \)

Problem ②: \( 15x^{12} - 35x^4 + 20x^2 + x \)

Step1: Find GCF of coefficients

Factors of \( 15 \): \( 1, 3, 5, 15 \)
Factors of \( 35 \): \( 1, 5, 7, 35 \)
Factors of \( 20 \): \( 1, 2, 4, 5, 10, 20 \)
Factor of \( 1 \): \( 1 \)
GCF of \( 15, 35, 20, 1 \) is \( 1 \).

Step2: Find GCF of variables

Lowest power of \( x \) is \( x^1 \) (from \( x \)).
GCF of variables: \( x \).

Step3: Factor out GCF

\( 15x^{12} - 35x^4 + 20x^2 + x = x(15x^{11} - 35x^3 + 20x + 1) \)

Problem ③: \( 39x^3 + 21x \)

Step1: Find GCF of coefficients

Factors of \( 39 \): \( 1, 3, 13, 39 \)
Factors of \( 21 \): \( 1, 3, 7, 21 \)
GCF of \( 39, 21 \) is \( 3 \).

Step2: Find GCF of variables

Lowest power of \( x \) is \( x^1 \).
GCF of variables: \( x \).

Step3: Factor out GCF

\( 39x^3 + 21x = 3x(13x^2 + 7) \)

Problem ④: \( 54x^{12} - 40x^2 \)

Answer:

\( 18xy(2x^3y^2 - 3xy + 5) \)