Question

find the greatest common factor of the expressions.
-16x²y⁹, 48xy⁹

Explanation:

Step1: Find GCF of coefficients

The coefficients are -16 and 48. The GCF of 16 and 48 is 16 (since \(48\div16 = 3\) and \(16\div16 = 1\)). Considering the sign, we take 16 (as GCF can be positive or negative, but we take the positive common factor here for simplicity, and we can adjust the sign later if needed, but in this case, the common factor with the highest absolute value is 16).

Step2: Find GCF of \(x\) terms

For \(x^2\) and \(x\), the GCF is \(x\) (since the lowest power of \(x\) is 1).

Step3: Find GCF of \(y\) terms

For \(y^9\) and \(y^9\), the GCF is \(y^9\) (since the power of \(y\) is 9 in both).

Step4: Combine the factors

Multiply the GCF of coefficients, \(x\) terms, and \(y\) terms: \(16\times x\times y^9 = 16xy^9\). But we also consider the sign. The first term is negative, but the GCF can be negative or positive. However, since 16 is the GCF of the absolute values, and we can have -16 as well? Wait, no. Wait, the GCF of -16 and 48: the factors of -16 are \(\pm1, \pm2, \pm4, \pm8, \pm16\), factors of 48 are \(\pm1, \pm2, \pm3, \pm4, \pm6, \pm8, \pm12, \pm16, \pm24, \pm48\). The greatest common factor (in terms of absolute value) is 16. But when we take the GCF of two numbers, one negative and one positive, the GCF is the greatest positive integer that divides both. So the GCF of -16 and 48 is 16. But wait, actually, the GCF of two integers \(a\) and \(b\) is the largest positive integer that divides both \(a\) and \(b\). So even if \(a\) is negative, the GCF is positive. But in the context of polynomials, sometimes we can take the negative GCF as well, but in this case, let's check:

If we take -16, does -16 divide 48? \(48\div(-16)= -3\), which is an integer. And -16 divides -16. So actually, the GCF can be -16 or 16? Wait, no, the definition of GCF for integers is the largest positive integer that divides both. But in the context of algebraic expressions, we can consider the GCF with the highest absolute value, regardless of sign. Wait, let's re - evaluate.

The factors of -16: \(\pm1, \pm2, \pm4, \pm8, \pm16\)

The factors of 48: \(\pm1, \pm2, \pm3, \pm4, \pm6, \pm8, \pm12, \pm16, \pm24, \pm48\)

The common factors are \(\pm1, \pm2, \pm4, \pm8, \pm16\)

The greatest common factor (the one with the largest absolute value) is 16 (positive) or -16 (negative). But when we look at the variables, let's see:

For the \(x\) terms, the lowest power is \(x^1\), for \(y\) terms, \(y^9\), and for coefficients, the GCF of -16 and 48 is 16 (since 16 is the largest positive integer that divides both -16 and 48). Wait, but -16 also divides both: -16 divides -16 (quotient 1) and -16 divides 48 (quotient -3). So actually, the GCF can be -16 or 16? But in the context of algebraic expressions, we usually take the GCF with the positive coefficient for the leading term, but let's check the two expressions:

\(-16x^2y^9=-16xy^9\times x\)

\(48xy^9 = 16xy^9\times3\)

So the common factor is \(16xy^9\) (since -16xy^9 also divides both, but 16xy^9 has a positive coefficient and is the GCF with the highest absolute value when considering the positive factor. Wait, no, the GCF is the greatest common factor, so we need to see which is larger in absolute value. 16 and -16 have the same absolute value. But in the case of the variables, the common factor is \(xy^9\) and the common factor of the coefficients (in terms of absolute value) is 16. So combining them, the GCF is \(16xy^9\) (or -16xy^9, but since 16 is the GCF of the absolute values, we take 16xy^9 as the GCF with the positive coefficient, which is the standard way unless specified otherwise).

Wait, let's verify:

Divide \(-16x^2y^9\) by \(16xy^9\): we get \(-x\), which is an integer (in terms of polynomials, it's a monomial, so it divides).

Divide \(48xy^9\) by \(16xy^9\): we get 3, which is an integer.

If we take -16xy^9:

Divide \(-16x^2y^9\) by -16xy^9: we get \(x\), which is good.

Divide \(48xy^9\) by -16xy^9: we get -3, which is also good.

But the greatest common factor (in terms of the largest absolute value) is 16xy^9 (since 16 and -16 have the same absolute value, but when we talk about GCF of polynomials, we usually take the one with the positive coefficient for the leading term, or the one with the highest absolute value. Since 16 is positive and has the same absolute value as -16, we can take 16xy^9 as the GCF.

Answer:

\(16xy^9\)