Question

directions: factor each polynomial. look for a gcf first.

1. $3p^2 - 75$
2. $5x^2 - 80$
3. $7n^2 - 7$
4. $16x^3 - 100x$
5. $3x^2 - 3y^2$
6. $8a^2 - 18b^2$
7. $4r^2 - 144$
8. $81m^4 - m^2$
9. $3k^4 - 48k^2$
10. $2x^3 - 200x$
11. $a^3 - a$
12. $32x^2 - 18y^2$

© gina wilson (all th

Answer:

Let's solve each polynomial by factoring out the GCF first and then using difference of squares if possible.

Problem 13: \( 3p^2 - 75 \)

Step 1: Factor out GCF

The GCF of \( 3p^2 \) and \( 75 \) is \( 3 \).
\( 3p^2 - 75 = 3(p^2 - 25) \)

Step 2: Factor difference of squares

\( p^2 - 25 \) is a difference of squares (\( p^2 - 5^2 \)), so we use \( a^2 - b^2 = (a - b)(a + b) \).
\( 3(p^2 - 25) = 3(p - 5)(p + 5) \)

Problem 14: \( 5x^2 - 80 \)

Step 1: Factor out GCF

The GCF of \( 5x^2 \) and \( 80 \) is \( 5 \).
\( 5x^2 - 80 = 5(x^2 - 16) \)

Step 2: Factor difference of squares

\( x^2 - 16 \) is a difference of squares (\( x^2 - 4^2 \)).
\( 5(x^2 - 16) = 5(x - 4)(x + 4) \)

Problem 15: \( 7n^2 - 7 \)

Step 1: Factor out GCF

The GCF of \( 7n^2 \) and \( 7 \) is \( 7 \).
\( 7n^2 - 7 = 7(n^2 - 1) \)

Step 2: Factor difference of squares

\( n^2 - 1 \) is a difference of squares (\( n^2 - 1^2 \)).
\( 7(n^2 - 1) = 7(n - 1)(n + 1) \)

Problem 16: \( 16x^3 - 100x \)

Step 1: Factor out GCF

The GCF of \( 16x^3 \) and \( 100x \) is \( 4x \).
\( 16x^3 - 100x = 4x(4x^2 - 25) \)

Step 2: Factor difference of squares

\( 4x^2 - 25 \) is a difference of squares (\( (2x)^2 - 5^2 \)).
\( 4x(4x^2 - 25) = 4x(2x - 5)(2x + 5) \)

Problem 17: \( 3x^2 - 3y^2 \)

Step 1: Factor out GCF

The GCF of \( 3x^2 \) and \( 3y^2 \) is \( 3 \).
\( 3x^2 - 3y^2 = 3(x^2 - y^2) \)

Step 2: Factor difference of squares

\( x^2 - y^2 \) is a difference of squares.
\( 3(x^2 - y^2) = 3(x - y)(x + y) \)

Problem 18: \( 8a^2 - 18b^2 \)

Step 1: Factor out GCF

The GCF of \( 8a^2 \) and \( 18b^2 \) is \( 2 \).
\( 8a^2 - 18b^2 = 2(4a^2 - 9b^2) \)

Step 2: Factor difference of squares

\( 4a^2 - 9b^2 \) is a difference of squares (\( (2a)^2 - (3b)^2 \)).
\( 2(4a^2 - 9b^2) = 2(2a - 3b)(2a + 3b) \)

Problem 19: \( 4r^2 - 144 \)

Step 1: Factor out GCF

The GCF of \( 4r^2 \) and \( 144 \) is \( 4 \).
\( 4r^2 - 144 = 4(r^2 - 36) \)

Step 2: Factor difference of squares

\( r^2 - 36 \) is a difference of squares (\( r^2 - 6^2 \)).
\( 4(r^2 - 36) = 4(r - 6)(r + 6) \)

Problem 20: \( 81m^4 - m^2 \)

Step 1: Factor out GCF

The GCF of \( 81m^4 \) and \( m^2 \) is \( m^2 \).
\( 81m^4 - m^2 = m^2(81m^2 - 1) \)

Step 2: Factor difference of squares

\( 81m^2 - 1 \) is a difference of squares (\( (9m)^2 - 1^2 \)).
\( m^2(81m^2 - 1) = m^2(9m - 1)(9m + 1) \)

Problem 21: \( 3k^4 - 48k^2 \)

Step 1: Factor out GCF

The GCF of \( 3k^4 \) and \( 48k^2 \) is \( 3k^2 \).
\( 3k^4 - 48k^2 = 3k^2(k^2 - 16) \)

Step 2: Factor difference of squares

\( k^2 - 16 \) is a difference of squares (\( k^2 - 4^2 \)).
\( 3k^2(k^2 - 16) = 3k^2(k - 4)(k + 4) \)

Problem 22: \( 2x^3 - 200x \)

Step 1: Factor out GCF

The GCF of \( 2x^3 \) and \( 200x \) is \( 2x \).
\( 2x^3 - 200x = 2x(x^2 - 100) \)

Step 2: Factor difference of squares

\( x^2 - 100 \) is a difference of squares (\( x^2 - 10^2 \)).
\( 2x(x^2 - 100) = 2x(x - 10)(x + 10) \)

Problem 23: \( a^3 - a \)

Step 1: Factor out GCF

The GCF of \( a^3 \) and \( a \) is \( a \).
\( a^3 - a = a(a^2 - 1) \)

Step 2: Factor difference of squares

\( a^2 - 1 \) is a difference of squares (\( a^2 - 1^2 \)).
\( a(a^2 - 1) = a(a - 1)(a + 1) \)

Problem 24: \( 32x^2 - 18y^2 \)

Step 1: Factor out GCF

The GCF of \( 32x^2 \) and \( 18y^2 \) is \( 2 \).
\( 32x^2 - 18y^2 = 2(16x^2 - 9y^2) \)

Step 2: Factor difference of squares

\( 16x^2 - 9y^2 \) is a difference of squares (\( (4x)^2 - (3y)^2 \)).
\( 2(16x^2 - 9y^2) = 2(4x - 3y)(4x + 3y) \)

Final Answers:

1. \( \boldsymbol{3(p - 5)(p + 5)} \)
2. \( \boldsymbol{5(x - 4)(x + 4)} \)
3. \( \boldsymbol{7(n - 1)(n + 1)} \)
4. \( \boldsymbol{4x(2x - 5)(2x + 5)} \)
5. \( \boldsymbol{3(x - y)(x + y)} \)
6. \( \boldsymbol{2(2a - 3b)(2a + 3b)} \)
7. \( \boldsymbol{4(r - 6)(r + 6)} \)
8. \( \boldsymbol{m^2(9m - 1)(9m + 1)} \)
9. \( \boldsymbol{3k^2(k - 4)(k + 4)} \)
10. \( \boldsymbol{2x(x - 10)(x + 10)} \)
11. \( \boldsymbol{a(a - 1)(a + 1)} \)
12. \( \boldsymbol{2(4x - 3y)(4x + 3y)} \)