FACTORING POLYNOMIAL EXPRESSIONS EXAM

Score: / 39

Instructions: Show your work for each question.

1. Write the prime factorization of [tex]18 x^4 y^2[/tex] and [tex]24 x^3 y^5[/tex], and determine the greatest common factor of the two monomials. (2 marks)

2. In each case, state the greatest common factor of the following sets of monomials. (1 mark each)

a) [tex]7 m, 14 m[/tex]

b) [tex]-10 x^5 z^6, -15 x^5 z^4[/tex]

c) [tex]8 a b^2, 9 a b, 6 a^2 b[/tex]

d) [tex]-28 p q r^3, -56 p^2 q, -64 q^2 r[/tex]

3. Identify each missing factor. (1 mark each)

a) [tex]12 a + 24 b = ( \, \, \, \, )(a + 2 b)[/tex]

b) [tex]4 x^2 y + 8 x^3 y^2 = \left(4 x^2 y\right)( \, \, \, \, )[/tex]

4. Factor the following polynomials by removing the greatest common factor. (1 mark each)

a) [tex]7 x y^2 + 49[/tex]

b) [tex]9 a b - 12 a c[/tex]

c) [tex]-5 x^2 - 10 x y - 20 x z[/tex]

d) [tex]14 a^2 b^2 + 21 a^3 b^2 - 35 a^2 b^3[/tex]

Sure, let's solve each part of the question step by step.

1. Find the prime factorization and greatest common factor.

- For the monomial [tex]\(18x^4y^2\)[/tex]:
- Prime factorization of 18: [tex]\(18 = 2 \times 3^2\)[/tex]
- So, [tex]\(18x^4y^2 = 2 \times 3^2 \times x^4 \times y^2\)[/tex]

- For the monomial [tex]\(24x^3y^5\)[/tex]:
- Prime factorization of 24: [tex]\(24 = 2^3 \times 3\)[/tex]
- So, [tex]\(24x^3y^5 = 2^3 \times 3 \times x^3 \times y^5\)[/tex]

- Determine the greatest common factor (GCF):
- For numbers: minimum powers of each prime factor
- [tex]\(2^1\)[/tex] (because [tex]\(2^1\)[/tex] in 18 and [tex]\(2^3\)[/tex] in 24)
- [tex]\(3^1\)[/tex] (because [tex]\(3^2\)[/tex] in 18 and [tex]\(3^1\)[/tex] in 24)
- For variables:
- [tex]\(x^3\)[/tex] (lowest power between [tex]\(x^4\)[/tex] and [tex]\(x^3\)[/tex])
- [tex]\(y^2\)[/tex] (lowest power between [tex]\(y^2\)[/tex] and [tex]\(y^5\)[/tex])

- So, GCF is: [tex]\(2 \times 3 \times x^3 \times y^2 = 6x^3y^2\)[/tex]

2. Find the GCF for each set of monomials.

a) [tex]\(7m, 14m\)[/tex]
- GCF: 7 (the common factor of 7 and 14) and [tex]\(m\)[/tex], so [tex]\(7m\)[/tex].

b) [tex]\(-10x^5z^6, -15x^5z^4\)[/tex]
- Numerical GCF: 5 (common factor of 10 and 15)
- Variable GCF: [tex]\(x^5\)[/tex] and [tex]\(z^4\)[/tex] (lowest power for z)
- So, GCF is [tex]\(-5x^5z^4\)[/tex].

c) [tex]\(8ab^2, 9ab, 6a^2b\)[/tex]
- There is no common numerical factor for 8, 9, and 6.
- Variable GCF: [tex]\(ab\)[/tex] (lowest power of [tex]\(a\)[/tex] and [tex]\(b\)[/tex])
- So, GCF is [tex]\(ab\)[/tex].

d) [tex]\(-28pqr^3, -56p^2q, -64q^2r\)[/tex]
- Numerical GCF: 4 (common factor of 28, 56, and 64)
- Variable GCF: [tex]\(q\)[/tex] (present in all terms)
- So, GCF is [tex]\(-4q\)[/tex].

3. Identify each missing factor.

a) [tex]\(12a + 24b\)[/tex]
- Factor out the GCF of 12 and 24, which is 12.
- Rewrite: [tex]\(12(a + 2b)\)[/tex], missing factor is [tex]\(12\)[/tex].

b) [tex]\(4x^2y + 8x^3y^2\)[/tex]
- Factor out the GCF: [tex]\(4x^2y\)[/tex].
- Rewrite: [tex]\(4x^2y(1 + 2xy)\)[/tex], missing factor is [tex]\((1 + 2xy)\)[/tex].

4. Factor each polynomial by removing the GCF.

a) [tex]\(7xy^2 + 49\)[/tex]
- Factor out 7: [tex]\(7(xy^2 + 7)\)[/tex].

b) [tex]\(9ab - 12ac\)[/tex]
- Factor out 3a: [tex]\(3a(3b - 4c)\)[/tex].

c) [tex]\(-5x^2 - 10xy - 20xz\)[/tex]
- Factor out [tex]\(-5x\)[/tex]: [tex]\(-5x(x + 2y + 4z)\)[/tex].

d) [tex]\(14a^2b^2 + 21a^3b^2 - 35a^2b^3\)[/tex]
- Factor out [tex]\(7a^2b^2\)[/tex]: [tex]\(7a^2b^2(2 + 3a - 5b)\)[/tex].

I hope this helps! Let me know if you have any questions!