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).
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

1. How is factoring a polynomial similar to factoring integers?
- Both processes break a number or expression into pieces that multiply together.
- For integers you find factors (like 12 = 3 × 4); for polynomials you find polynomial factors (like 12x + 18 = 6(2x + 3)).
- In both cases you often pull out the greatest common factor (GCF) first.

2. Why does the GCF of the variables of a polynomial have the least exponent of any variable term in the polynomial?
- The GCF must divide every term. If one term has x^3 and another has x^5, x^5 cannot divide x^3, but x^3 divides x^5. So you must take the smaller (least) exponent for each variable. In other words, use the minimum exponent among the terms so the factor divides every term.

4. Andrew factored 3x^2y - 6xy^2 + 3xy as 3xy(x - 2y).
Find the GCF of each pair of monomials.
- All three monomials (3x^2y, -6xy^2, 3xy) share the same GCF: 3xy.
- So the GCF of any pair from those three is also 3xy.

7. GCF of x^3y^2 and x^5y
- x^min(3,5) y^min(2,1) = x^3y

9. GCF of 4x^3 and 9y^5
- Coefficients 4 and 9 have gcd 1 and the variables are different, so GCF = 1

10. GCF of 12a^5b and 16a^4b^2
- 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 the step-by-step division for any factoring above.