Question

1. what is the greatest common factor of the terms of the polynomial $-25y^3 + 15y^2 - 5y$?

Explanation:

Step1: Analyze coefficients

Find GCF of -25, 15, -5. Factors of 25: 1,5,25; 15:1,3,5,15; 5:1,5. GCF is 5 (consider sign, -5 is also common, but GCF is positive max, so 5).

Step2: Analyze variables

For \( y^3, y^2, y \), the least power of \( y \) is \( y^1 \) (i.e., \( y \)).

Step3: Combine

GCF of coefficients (5) and variables (\( y \)) gives \( 5y \)? Wait, no: Wait, -25,15,-5: GCF of absolute values 25,15,5 is 5. The sign: all terms have negative or positive, but GCF can be negative? Wait, actually, GCF of -25,15,-5: let's see, -5 divides -25 ( -25 / -5 =5), 15 / -5 = -3, -5 / -5=1. So GCF can be -5y? Wait no, let's recalculate.

Wait, coefficients: -25, 15, -5. Let's factor each:

-25 = -1 * 5 * 5

15 = 3 * 5

-5 = -1 * 5

So common factors: 5 (and -5? Let's check: -5: -25 / -5 =5, 15 / -5 = -3, -5 / -5=1. So both 5 and -5 are common. But usually, GCF is taken as the greatest common factor in absolute value, but sometimes with the sign that matches the first term. Wait, the polynomial is -25y³ +15y² -5y. The first term is negative, so maybe GCF is -5y? Wait no, let's check:

If we take 5y: -25y³ /5y = -5y², 15y² /5y=3y, -5y /5y= -1. So that works. If we take -5y: -25y³ /-5y=5y², 15y² /-5y= -3y, -5y /-5y=1. Also works. But which is the greatest? Wait, GCF is the greatest common factor, so in terms of absolute value, 5y and -5y have same absolute value. But maybe the problem expects the positive GCF? Wait, no, let's check the terms:

Wait, the polynomial is -25y³ +15y² -5y. Let's factor out -5y: -5y(5y² -3y +1). Or factor out 5y: 5y(-5y² +3y -1). Both are correct, but which is the GCF? Wait, actually, the GCF of the terms is the largest expression that divides all terms. Let's check each term:

-25y³ ÷ (-5y) = 5y²

15y² ÷ (-5y) = -3y

-5y ÷ (-5y) = 1

All integers (or monomials). Now check 5y:

-25y³ ÷5y = -5y²

15y² ÷5y=3y

-5y ÷5y= -1

Also integers. So both -5y and 5y are common factors. But which is the greatest? Wait, in terms of the greatest common factor, the GCF is the largest (in absolute value) common factor. The absolute value of 5y and -5y is same. But maybe the problem considers the positive GCF? Wait, no, let's see the original terms: the first term is -25y³, so if we take GCF as -5y, then factoring gives -5y(5y² -3y +1), which is correct. But let's check the exponents again. The variables: y³, y², y. The lowest power is y^1, so y. Coefficients: GCF of -25,15,-5. Let's compute GCF of -25,15,-5. The GCF of a set of integers is the largest integer that divides all of them. So for -25,15,-5:

Divisors of -25: ±1, ±5, ±25

Divisors of 15: ±1, ±3, ±5, ±15

Divisors of -5: ±1, ±5

Common divisors: ±1, ±5

The greatest (in absolute value) is 5 (or -5, same absolute value). So GCF of coefficients is 5 (or -5), and GCF of variables is y. So combining, GCF is 5y or -5y. But let's check the polynomial: -25y³ +15y² -5y. Let's factor out -5y: -5y(5y² - 3y + 1). Let's factor out 5y: 5y(-5y² + 3y - 1). Both are correct, but which is the GCF? Wait, actually, the GCF can be negative, but typically, we take the GCF with the sign that makes the first term in the parentheses positive? Wait, no, the question is "greatest common factor of the terms", so the terms are -25y³, 15y², -5y. Let's find the GCF of each term:

-25y³ = -5 * 5 * y * y * y

15y² = 3 * 5 * y * y

-5y = -1 * 5 * y

So the common factors are 5 and y (since each term has at least one 5 and at least one y). Wait, -25y³ has 5*5*y*y*y, 15y² has 5*3*y*y, -5y has 5*(-1)*y. So the common prime factors are 5 (once) and y (once). So GCF is 5*y = 5y? Wait, but -25y³ /5y = -5y², 15y² /5y=3y, -5y /5y= -1. So that works. So the GCF is 5y? Wait, but earlier I thought -5y, but no, 5y divides all terms: -25y³ ÷5y = -5y² (integer coefficient), 15y² ÷5y=3y (integer), -5y ÷5y= -1 (integer). So yes, 5y is a common factor. Is there a larger one? Let's see, 25y? -25y³ ÷25y= -y², 15y² ÷25y= 0.6y (not integer), so no. 5y²? -25y³ ÷5y²= -5y, 15y² ÷5y²=3, -5y ÷5y²= -1/y (not integer), so no. So the GCF is 5y? Wait, but wait, the coefficient of the first term is -25, so maybe the GCF is -5y? Let's check: -25y³ ÷-5y=5y², 15y² ÷-5y= -3y, -5y ÷-5y=1. All integers. So both 5y and -5y are common factors. But which is the greatest? In terms of absolute value, they are equal. But maybe the problem expects the positive one? Wait, no, let's check the definition: The greatest common factor (GCF) of a set of numbers (or terms) is the largest positive integer (or monomial) that divides each of the numbers (or terms) without leaving a remainder. So "largest positive", so we take the positive GCF. So 5y? Wait, but wait, the terms have negative coefficients, but GCF is positive. Wait, no, the GCF can be negative, but the "greatest" in GCF refers to the greatest in absolute value, but since we are dealing with integers, the GCF is the largest integer (positive) that divides all, but for monomials, it's the monomial with the largest coefficient (in absolute value) and highest power of variables that divides all. Wait, maybe I made a mistake earlier. Let's re-express each term:

Term 1: -25y³ = (-5y) * (5y²)

Term 2: 15y² = (-5y) * (-3y)

Term 3: -5y = (-5y) * (1)

Ah! So here, -5y is a factor of all three terms. Let's check:

-25y³ ÷ (-5y) = 5y² (correct)

15y² ÷ (-5y) = -3y (correct)

-5y ÷ (-5y) = 1 (correct)

Now, is there a larger factor? Let's see, if we take -10y: -25y³ ÷-10y=2.5y² (not integer), so no. -5y is a factor, 5y is also a factor. But which is the GCF? Wait, the GCF of the terms -25y³, 15y², -5y. Let's list the factors of each term:

Factors of -25y³: all monomials of the form \( k y^n \) where \( k \) divides -25 and \( n \leq 3 \).

Factors of 15y²: \( k y^n \) where \( k \) divides 15 and \( n \leq 2 \).

Factors of -5y: \( k y^n \) where \( k \) divides -5 and \( n \leq 1 \).

Common factors must have \( k \) dividing -25,15,-5 (so \( k \) divides GCF of -25,15,-5 which is 5 or -5) and \( n \leq 1 \) (since the least power of y is 1). So possible common factors: \( \pm1y, \pm5y \).

Now, "greatest" common factor: in terms of the coefficient's absolute value, 5y and -5y have the same absolute value (5), which is greater than 1. So between 5y and -5y, which is greater? In terms of monomials, we can consider the coefficient. If we take the coefficient's absolute value, 5 is greater than -5? No, absolute value is same. Wait, maybe the problem considers the GCF as the one with the sign that is common to all terms? The first term is negative, the second is positive, the third is negative. So no common sign. So we can take the positive GCF or negative. But let's check the original problem: the polynomial is -25y³ +15y² -5y. Let's factor out -5y: -5y(5y² - 3y + 1). Let's factor out 5y: 5y(-5y² + 3y - 1). Both are correct, but which is the GCF? Wait, actually, the GCF of the terms is the largest monomial that divides each term. Let's check the coefficients:

-25, 15, -5. The GCF of these integers (using the definition of GCF for integers: the largest integer that divides all) is 5? Wait, no: GCF of -25,15,-5. Let's compute GCF of 25,15,5 (absolute values) is 5. Then, since the signs: two negative, one positive. The GCF can be negative or positive, but typically, we take the positive GCF for integers, but for monomials, it's a bit different. Wait, maybe I was overcomplicating. Let's do it step by step:

1. Find GCF of coefficients: -25, 15, -5.

- Prime factors:

-25: -1 × 5 × 5

15: 3 × 5

-5: -1 × 5

- Common prime factor: 5 (with multiplicity 1)

- So GCF of coefficients (considering sign, but GCF for integers is positive, but since we can have negative, but let's see: 5 divides -25 ( -25 ÷5 = -5), 15 ÷5=3, -5 ÷5= -1. So 5 is a common factor. -5 also divides: -25 ÷-5=5, 15 ÷-5= -3, -5 ÷-5=1. So both 5 and -5 are common factors.

2. Find GCF of variables: \( y^3, y^2, y \).

- The least power of \( y \) is \( y^1 \) (i.e., \( y \)).

3. Combine GCF of coefficients and variables:

- If we take GCF of coefficients as 5, then GCF is \( 5y \).

- If we take GCF of coefficients as -5, then GCF is \( -5y \).

But which is the "greatest" common factor? In terms of the monomial's "size", we can consider the coefficient's absolute value. Since |5| = |-5|, both are valid, but let's check the original polynomial. When we factor out \( -5y \), the first term in the parentheses becomes positive (5y²), which is a common practice (to have the leading coefficient in the parentheses positive). Let's verify:

- Factoring out \( -5y \): \( -5y(5y² - 3y + 1) \).

- Factoring out \( 5y \): \( 5y(-5y² + 3y - 1) \).

Both are correct, but the first one has a positive leading coefficient in the parentheses, which is preferred. However, the question is about the GCF of the terms, not about factoring the polynomial. So we need to find the GCF of the terms \( -25y³ \), \( 15y² \), \( -5y \).

Let's check each term divided by \( -5y \):

- \( -25y³ ÷ -5y = 5y² \) (integer)

- \( 15y² ÷ -5y = -3y \) (integer)

- \( -5y ÷ -5y = 1 \) (integer)

Now check divided by \( 5y \):

- \( -25y³ ÷ 5y = -5y² \) (integer)

- \( 15y² ÷ 5y = 3y \) (integer)

- \( -5y ÷ 5y = -1 \) (integer)

So both \( -5y \) and \( 5y \) are common factors. But which is the greatest? In terms of the coefficient's absolute value, they are equal. However, in the context of GCF for monomials, we usually take the GCF with the sign that is common to the majority of terms or the one that makes the factoring simpler, but strictly speaking, the GCF can be either. Wait, but let's check the definition: The greatest common factor of a set of monomials is the product of the GCF of their coefficients and the GCF of their variable parts.

GCF of coefficients (-25,15,-5): as integers, the GCF is 5 (since GCF of integers is the largest positive integer that divides all, but actually, GCF can be negative, but in standard definitions, GCF is positive. Wait, no, the GCF of integers is defined as the largest integer (positive) that divides all. So GCF of -25,15,-5 is 5. Then GCF of variables is \( y \). So GCF is \( 5y \).

Wait, maybe I was wrong earlier about the sign. Let's confirm with an example: GCF of -4, 6, -2. The GCF of 4,6,2 is 2. So GCF of -4,6,-2 is 2. Then, applying that, GCF of -25,15,-5 is 5. So then GCF of variables is \( y \), so GCF is \( 5y \).

Let's check:

-25y³ ÷5y = -5y² (integer)

15y² ÷5y=3y (integer)

-5y ÷5y= -1 (integer)

Yes, that works. So the GCF is \( 5y \)? Wait, but wait, -5y also works, but according to the definition of GCF for integers being positive, the GCF of the coefficients is 5, so the GCF of the terms is \( 5y \). Wait, no, the terms have coefficients -25,15,-5, so their GCF (as integers) is 5, so the GCF of the terms (monomials) is 5y.

Wait, I think I made a mistake earlier with the sign. The GCF of the coefficients (as integers) is 5 (positive), so the GCF of the monomials is 5y.

Answer:

\boxed{5y}? Wait, no, wait: Wait, let's check again. Wait, -25y³, 15y², -5y. Let's factor each:

-25y³ = (-5y)(5y²)

15y² = (-5y)(-3y)