Question

directions: find the gcf of each.

1. (21m^3, 28m)
2. (8x^2y, -12x^4y^2)
3. (13x, 26)

Explanation:

Problem 1: Find the GCF of \(21m^2\) and \(28m\)

Step 1: Factor the coefficients

Factor \(21\) and \(28\).
\(21 = 3\times7\)
\(28 = 4\times7=2^2\times7\)
The common factor of \(21\) and \(28\) is \(7\).

Step 2: Factor the variables

For the variable \(m\), in \(21m^2\) the exponent of \(m\) is \(2\) (\(m^2=m\times m\)) and in \(28m\) the exponent of \(m\) is \(1\) (\(m = m\)). The lowest power of \(m\) is \(1\), so the common variable factor is \(m\).

Step 3: Combine the common factors

Multiply the common coefficient factor and the common variable factor.
The GCF is \(7\times m=7m\)

Problem 2: Find the GCF of \(8x^2y\) and \(- 12x^3y^2\)

Step 1: Factor the coefficients

Factor \(8\) and \(- 12\).
\(8=2^3\)
\(-12=-1\times2^2\times3\)
The common factor of \(8\) and \(12\) (we consider the absolute values for GCF of coefficients) is \(2^2 = 4\)

Step 2: Factor the variables

For the variable \(x\): In \(8x^2y\) the exponent of \(x\) is \(2\) (\(x^2=x\times x\)) and in \(-12x^3y^2\) the exponent of \(x\) is \(3\) (\(x^3=x\times x\times x\)). The lowest power of \(x\) is \(2\), so the common factor for \(x\) is \(x^2\)

For the variable \(y\): In \(8x^2y\) the exponent of \(y\) is \(1\) (\(y = y\)) and in \(-12x^3y^2\) the exponent of \(y\) is \(2\) (\(y^2=y\times y\)). The lowest power of \(y\) is \(1\), so the common factor for \(y\) is \(y\)

Step 3: Combine the common factors

Multiply the common coefficient factor, common \(x\) factor and common \(y\) factor.
The GCF is \(4\times x^2\times y = 4x^2y\)

Problem 3: Find the GCF of \(13x\) and \(26\)

Step 1: Factor the coefficients

Factor \(13\) and \(26\)
\(13 = 13\)
\(26=2\times13\)
The common factor of \(13\) and \(26\) is \(13\)

Step 2: Factor the variables

In \(13x\) we have the variable \(x\) with exponent \(1\) and in \(26\) there is no variable. So there is no common variable factor.

Step 3: Combine the common factors

The GCF is the common coefficient factor (since there is no common variable factor). So the GCF is \(13\)

Answer:

s:

1. The GCF of \(21m^2\) and \(28m\) is \(\boldsymbol{7m}\)
2. The GCF of \(8x^2y\) and \(-12x^3y^2\) is \(\boldsymbol{4x^2y}\)
3. The GCF of \(13x\) and \(26\) is \(\boldsymbol{13}\)