Question

③ 39x³ + 21x
④ 54x¹³ - 40x²
⑤ 80x⁶y² - 24xy

Explanation:

(for each factoring):

• For \(39x^{3}+21x\), we identify the greatest common factor (GCF) of the terms (3x) and factor it out, leaving the remaining polynomial \(13x^{2}+7\) inside the parentheses.
• For \(54x^{3}-40x^{2}\) (corrected from the likely typo), we find the GCF (2x²) and factor it out, resulting in \(2x^{2}(27x - 20)\).
• For \(80x^{6}y^{2}-24xy\), we determine the GCF (8xy) by analyzing coefficients and variable powers, then factor it out to get \(8xy(10x^{5}y - 3)\).

Answer:

(for factoring each expression):

3. \(39x^{3}+21x\)

Step1: Find GCF of coefficients and variables

• Coefficients: GCF of 39 and 21. Prime factors of 39: \(3\times13\), prime factors of 21: \(3\times7\). So GCF is 3.
• Variables: For \(x^{3}\) and \(x\), the lowest power of \(x\) is \(x^{1}\) (i.e., \(x\)).
• GCF of the terms is \(3x\).

Step2: Factor out GCF

\(39x^{3}+21x = 3x(13x^{2}+7)\)

4. \(54x^{3}-40x^{2}\) (assuming the "13" is a typo, likely \(x^{3}\))

Step1: Find GCF of coefficients and variables

• Coefficients: GCF of 54 and 40. Prime factors of 54: \(2\times3^{3}\), prime factors of 40: \(2^{3}\times5\). So GCF is 2.
• Variables: For \(x^{3}\) and \(x^{2}\), the lowest power of \(x\) is \(x^{2}\).
• GCF of the terms is \(2x^{2}\).

Step2: Factor out GCF

\(54x^{3}-40x^{2}=2x^{2}(27x - 20)\)

5. \(80x^{6}y^{2}-24xy\)

Step1: Find GCF of coefficients and variables

• Coefficients: GCF of 80 and 24. Prime factors of 80: \(2^{4}\times5\), prime factors of 24: \(2^{3}\times3\). So GCF is \(2^{3}=8\).
• Variables: For \(x^{6}y^{2}\) and \(xy\), the lowest power of \(x\) is \(x^{1}\) (i.e., \(x\)) and lowest power of \(y\) is \(y^{1}\) (i.e., \(y\)). So variable part GCF is \(xy\).
• Overall GCF: \(8\times xy = 8xy\).

Step2: Factor out GCF

\(80x^{6}y^{2}-24xy=8xy(10x^{5}y - 3)\)