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**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]

Answer :

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!

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Rewritten by : Jeany

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