Factor the GCF from the polynomial.18x3y2−27x2y3+9x4y18x^3y^2-27x^2y^3+9x^4y

9 x 2 y ( 2 x y − 3 y 2 + x 2 ) 9x^2y\left(2xy-3y^2+x^2\right)

Factor the GCF from the polynomial. 18 x 3 y 2 − 27 x 2 y 3 + 9 x 4 y 18x^3y^2-27x^2y^3+9x^4y

Here, we've been asked to factor this long polynomial using the GCF method. So even though this problem has two different types of variable, we're gonna follow the same steps that we've been using before by first writing out the prime factorization for each of the terms in the polynomial. So for our first term we have 18 x cubed times y squared, and its prime factorization is two times three times three times three x's and times two y's. Then for our second term, 27 x squared y cubed, We have three times three times three times two x's and three y's. And for our last term, nine x to the fourth times y, We have three times three times four x's and one y. So what are the factors that these all have in common? Well, we've got two three's in each of them. We also have two x's in each of them, and we have one y. So our greatest common factor then is three times three, which is nine, x times x, which is x squared, times y. And now I'm going to rewrite each of the terms in the polynomial as a product with the greatest common factor. So 9x squared times y. What can I multiply that by to get 18 x cubed y squared? Well that'll be two times x times y. And what about the second term in this polynomial? How can I make 27 x squared y cubed? That'll be multiplying by three times y squared. And for the last term, nine x squared times y, we can multiply that by x squared to get nine x to the fourth times y. So I can put x squared here. Now my final step is to factor out the greatest common factor. So doing that, we're left with two x y plus three y squared plus x squared. And that is our final answer.

Factor the GCF from the polynomial. 18 x 3 y 2 − 27 x 2 y 3 + 9 x 4 y 18x^3y^2-27x^2y^3+9x^4y

9 x 2 y ( 2 x y − 3 y 2 + x 2 ) 9x^2y\left(2xy-3y^2+x^2\right)

9 x 2 y ( 2 x 2 − 3 x y + 3 x 2 ) 9x^2y\left(2x^2-3xy+3x^2\right)

9 x 2 y ( 2 x y + 3 y 2 + x 2 ) 9x^2y\left(2xy+3y^2+x^2\right)

9 x 2 y ( 2 x 2 − 3 x y + x 3 ) 9x^2y\left(2x^2-3xy+x^3\right)

7. Factoring

Factoring by Greatest Common Factor & Grouping

Intermediate Algebra

7. Factoring

Factoring by Greatest Common Factor & Grouping

Struggling with Intermediate Algebra?

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Multiple Choice

Factor the GCF from the polynomial.
$18 x^{3} y^{2} - 27 x^{2} y^{3} + 9 x^{4} y$

$9 x^{2} y (2 x^{2} - 3 x y + 3 x^{2})$

$9 x^{2} y (2 x y + 3 y^{2} + x^{2})$

$9 x^{2} y (2 x^{2} - 3 x y + x^{3})$

$9 x^{2} y (2 x y - 3 y^{2} + x^{2})$

Verified step by step guidance

Identify the greatest common factor (GCF) of the coefficients: Look at the numbers 18, 27, and 9. Determine the largest number that divides all three evenly.

Find the GCF of the variable parts by looking at the powers of x and y in each term: For x, find the smallest exponent among 3, 2, and 4; for y, find the smallest exponent among 2, 3, and 1.

Write the GCF as the product of the GCF of the coefficients and the variables with their smallest exponents. This will be the factor you take out of the polynomial.

Divide each term of the polynomial by the GCF you found. This will give you the simplified terms inside the parentheses after factoring out the GCF.

Express the original polynomial as the product of the GCF and the simplified polynomial inside parentheses, i.e., write it in the form $\text{GCF} \times (\text{simplified polynomial})$.

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