Question

gcf assignment #2

1. $35v^2 - 14v - 42$
2. $-24a^3 + 12a^2 + 30$
3. $10x^6 - 4x^5 + 4x^4$
4. $-24x^6 - 72x^2y^2 + 8x^2$
5. $27x^6y + 54x^5y^2 + 72x^5y$
6. $-45x + 15x^4 + 10xy$
7. $9 - 18xy + 3x^2y^3$
8. $10x^4 + 100x^2 + 10xy$
9. $30yx^3 + 100yx + 20y^2$

Answer:

Let's solve each problem by finding the greatest common factor (GCF) of the terms in each polynomial.

Problem 1: \( 35v^2 - 14v - 42 \)

Step 1: Find GCF of coefficients

The coefficients are 35, -14, -42.
Prime factors:

• \( 35 = 5 \times 7 \)
• \( 14 = 2 \times 7 \)
• \( 42 = 2 \times 3 \times 7 \)

GCF of coefficients: \( 7 \)

Step 2: Find GCF of variables

There is only \( v \) with the lowest power \( v^0 = 1 \) (since -42 has no variable).

Step 3: Factor out GCF

\( 35v^2 - 14v - 42 = 7(5v^2 - 2v - 6) \)

Problem 2: \( -24a^3 + 12a^2 + 30 \)

Step 1: Find GCF of coefficients

Coefficients: -24, 12, 30.
Prime factors:

• \( 24 = 2^3 \times 3 \)
• \( 12 = 2^2 \times 3 \)
• \( 30 = 2 \times 3 \times 5 \)

GCF of coefficients: \( 6 \) (we can factor out -6 to make the leading coefficient positive).

Step 2: Factor out GCF

\( -24a^3 + 12a^2 + 30 = -6(4a^3 - 2a^2 - 5) \)

Problem 3: \( 10x^6 - 4x^5 + 4x^4 \)

Step 1: Find GCF of coefficients

Coefficients: 10, -4, 4.
Prime factors:

• \( 10 = 2 \times 5 \)
• \( 4 = 2^2 \)

GCF of coefficients: \( 2 \)

Step 2: Find GCF of variables

Variables: \( x^6, x^5, x^4 \). Lowest power: \( x^4 \).

Step 3: Factor out GCF

\( 10x^6 - 4x^5 + 4x^4 = 2x^4(5x^2 - 2x + 2) \)

Problem 4: \( -24x^6 - 72x^2y^2 + 8x^2 \)

Step 1: Find GCF of coefficients

Coefficients: -24, -72, 8.
Prime factors:

• \( 24 = 2^3 \times 3 \)
• \( 72 = 2^3 \times 3^2 \)
• \( 8 = 2^3 \)

GCF of coefficients: \( 8 \) (factor out -8 to make leading term positive).

Step 2: Find GCF of variables

Variables: \( x^6, x^2y^2, x^2 \). Lowest power of \( x \): \( x^2 \); \( y \) has lowest power \( y^0 = 1 \).

Step 3: Factor out GCF

\( -24x^6 - 72x^2y^2 + 8x^2 = -8x^2(3x^4 + 9y^2 - 1) \)

Problem 5: \( 27x^6y + 54x^5y^2 + 72x^5y \)

Step 1: Find GCF of coefficients

Coefficients: 27, 54, 72.
Prime factors:

• \( 27 = 3^3 \)
• \( 54 = 2 \times 3^3 \)
• \( 72 = 2^3 \times 3^2 \)

GCF of coefficients: \( 9 \) (wait, 27, 54, 72: GCF is 9? Wait, 27=3³, 54=2×3³, 72=2³×3². GCF is 3²=9? Wait, no: 27, 54, 72. 27 divides 54 (54=2×27), 27 divides 72? 72÷27=2.666… No. Wait, 9: 27÷9=3, 54÷9=6, 72÷9=8. Yes. Wait, but 27, 54, 72: GCF is 9? Wait, 27=3³, 54=2×3³, 72=2³×3². The common prime factors are 3²=9. Correct.

Wait, actually, 27, 54, 72: GCF is 9? Wait, 27×2=54, 27×2.666=72. No. Wait, 9: 27÷9=3, 54÷9=6, 72÷9=8. Yes. But wait, 27, 54, 72: GCF is 9? Wait, 3: 27÷3=9, 54÷3=18, 72÷3=24. Then GCF of 9, 18, 24 is 3. So total GCF is 3×3=9? Wait, no. Let's do it properly:

Prime factors:

• 27: 3×3×3
• 54: 2×3×3×3
• 72: 2×2×2×3×3

Common factors: 3×3=9. Yes.

Step 2: Find GCF of variables

Variables: \( x^6y, x^5y^2, x^5y \).

• \( x \): lowest power \( x^5 \)
• \( y \): lowest power \( y^1 \)

Step 3: Factor out GCF

\( 27x^6y + 54x^5y^2 + 72x^5y = 9x^5y(3x + 6y + 8) \)

Problem 6: \( -45x + 15x^4 + 10xy \)

Step 1: Find GCF of coefficients

Coefficients: -45, 15, 10.
Prime factors:

• \( 45 = 3^2 \times 5 \)
• \( 15 = 3 \times 5 \)
• \( 10 = 2 \times 5 \)

GCF of coefficients: \( 5 \) (factor out -5 to make leading term positive).

Step 2: Find GCF of variables

Variables: \( x, x^4, xy \). Lowest power of \( x \): \( x^1 \); \( y \) has lowest power \( y^0 = 1 \).

Step 3: Factor out GCF

\( -45x + 15x^4 + 10xy = -5x(9 - 3x^3 - 2y) \)

Problem 7: \( 9 - 18xy + 3x^2y^3 \)

Step 1: Find GCF of coefficients

Coefficients: 9, -18, 3.
Prime factors:

• \( 9 = 3^2 \)
• \( 18 = 2 \times 3^2 \)
• \( 3 = 3 \)

GCF of coefficients: \( 3 \)

Step 2: Find GCF of variables

Variables: \( 1, xy, x^2y^3 \). Lowest power: \( x^0y^0 = 1 \).

Step 3: Factor out GCF

\( 9 - 18xy + 3x^2y^3 = 3(3 - 6xy + x^2y^3) \)

Problem 8: \( 10x^4 + 100x^2 + 10xy \)

Step 1: Find GCF of coefficients

Coefficients: 10, 100, 10.
Prime factors:

• \( 10 = 2 \times 5 \)
• \( 100 = 2^2 \times 5^2 \)

GCF of coefficients: \( 10 \)

Step 2: Find GCF of variables

Variables: \( x^4, x^2, xy \). Lowest power of \( x \): \( x^1 \); \( y \) has lowest power \( y^0 = 1 \).

Step 3: Factor out GCF

\( 10x^4 + 100x^2 + 10xy = 10x(x^3 + 10x + y) \)

Problem 9: \( 30yx^3 + 100yx + 20y^2 \)

Step 1: Find GCF of coefficients

Coefficients: 30, 100, 20.
Prime factors:

• \( 30 = 2 \times 3 \times 5 \)
• \( 100 = 2^2 \times 5^2 \)
• \( 20 = 2^2 \times 5 \)

GCF of coefficients: \( 10 \)

Step 2: Find GCF of variables

Variables: \( yx^3, yx, y^2 \). Lowest power of \( y \): \( y^1 \); \( x \) has lowest power \( x^0 = 1 \).

Step 3: Factor out GCF

\( 30yx^3 + 100yx + 20y^2 = 10y(3x^3 + 10x + 2y) \)

Final Answers:

1. \( \boldsymbol{7(5v^2 - 2v - 6)} \)
2. \( \boldsymbol{-6(4a^3 - 2a^2 - 5)} \)
3. \( \boldsymbol{2x^4(5x^2 - 2x + 2)} \)
4. \( \boldsymbol{-8x^2(3x^4 + 9y^2 - 1)} \)
5. \( \boldsymbol{9x^5y(3x + 6y + 8)} \)
6. \( \boldsymbol{-5x(9 - 3x^3 - 2y)} \)
7. \( \boldsymbol{3(3 - 6xy + x^2y^3)} \) (Wait, earlier I had a typo. Let's recheck Problem 7:

Original: \( 9 - 18xy + 3x^2y^3 \). GCF is 3. So \( 3(3 - 6xy + x^2y^3) \). Correct.

1. \( \boldsymbol{10x(x^3 + 10x + y)} \)
2. \( \boldsymbol{10y(3x^3 + 10x + 2y)} \)