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

Beginning & Intermediate Algebra

7. Factoring

Factoring by Greatest Common Factor & Grouping

Struggling with Beginning & 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: For each term, look at the powers of $x$ and $y$. Choose the smallest exponent of $x$ and the smallest exponent of $y$ that appear in all terms.

Write the GCF by combining the numerical GCF with the variables raised to their smallest powers. This will be the factor you take out of the polynomial.

Divide each term of the polynomial by the GCF you found. Write the polynomial as the product of the GCF and the resulting simplified expression inside parentheses.

Double-check the signs and terms inside the parentheses to ensure they match the original polynomial when multiplied back by the GCF.

5:47m

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