Question

in 9th grade terms
1.How is factoring a polynomial similar to factoring integers ?
2.Why does the GCF of the variables of a polynomial have the least exponent of any variable term in the polynomial?
4. Andrew factored 3x^2y - 6xy^2 + 3xy as 3xy(x - 2y). describe and correct the error

Find the GCF of each pair of monomials.
7. x^3y^2 and x^5y
9. 4x^3 and 9y^5
10. 12a^5b and 16a^4b^2
Factor out the GCF from each polynomial.
12. 10a^2b + 12ab^2
14. 15x^3y - 10x^2y^3
15. x^10 + x^9 - x^8

Answer 1

I’ll answer each part in simple 9th-grade terms.

1) How is factoring a polynomial similar to factoring integers?
- Both look for a greatest common factor (GCF) and “pull it out.”
- For integers you pull out the largest number that divides all terms. For polynomials you pull out the largest number and variable powers that divide every term.
- In both cases you use the distributive property: a(b + c) = ab + ac. Factoring is just reversing that: ab + ac = a(b + c).

2) Why does the GCF of the variables use the smallest exponent in the terms?
- A factor must divide every term. If one term has x^3 and another has x^5, x^4 does not divide x^3, so you cannot factor out x^4. The largest power of x that divides all terms is the smallest exponent among them. Example: between x^3 and x^5 the GCF for x is x^3.

4) Andrew’s factoring: 3x^2y - 6xy^2 + 3xy
- Andrew wrote 3xy(x - 2y). That misses the +3xy term’s contribution.
- Correct: factor out 3xy (the GCF):
3x^2y - 6xy^2 + 3xy = 3xy(x - 2y + 1) = 3xy(x + 1 - 2y).
- Error explained: He left out the +1 inside the parentheses. 3xy times 1 is the +3xy term.

Find the GCF of each pair of monomials:
7) x^3y^2 and x^5y
- GCF: x^3 y (choose the smaller exponent for each variable)

9) 4x^3 and 9y^5
- GCF: 1 (no common variable factors; 4 and 9 are relatively prime)

10) 12a^5b and 16a^4b^2
- Coefficient GCF: gcd(12,16) = 4
- a: min(5,4) = a^4
- b: min(1,2) = b
- GCF: 4a^4b

Factor out the GCF from each polynomial:
12) 10a^2b + 12ab^2
- GCF = 2ab
- Factor: 2ab(5a + 6b)

14) 15x^3y - 10x^2y^3
- GCF = 5x^2y
- Factor: 5x^2y(3x - 2y^2)

15) x^10 + x^9 - x^8
- GCF = x^8
- Factor: x^8(x^2 + x - 1)

If you want, I can show each factoring step in more detail.