How do you factor $a^{2} b + 6 a b^{2} + 9 b^{2}$ $a^2b+6ab^2+9b^2$ ?

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How do you factor $a^{2} b + 6 a b^{2} + 9 b^{2}$ $a^2b+6ab^2+9b^2$ ?

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Answer 1 · 2017-01-18T19:34:27

See solution below

Explanation:

To factor this, because all terms are positive and there are two variables we know the factoring will be in the form:

$(j a + k b) (l a + m b)$ $(color(red)(j)a + color(blue)(k)b)(color(green)(l)a + color(purple)(m)b)$

Where $j$ $color(red)(j)$ , $k$ $color(blue)(k)$ , $l$ $color(green)(l)$ and $m$ $color(purple)(m)$ are constants.

We also know because of the coefficients in the original problem:
$j \times l = 1$ $color(red)(j) xx color(green)(l) = 1$
$k \times m = 9$ $color(blue)(k) xx color(purple)(m) = 9$
$(j \times m) + (k \times l) = 6$ $(color(red)(j) xx color(purple)(m)) + (color(blue)(k) xx color(green)(l)) = 6$

"Playing" with the factors for $9$ $9$ (1x9, 3x3, 9x1) Gives:

$(1 a + 3 b) (1 a + 3 b)$ $(color(red)(1)a + color(blue)(3)b)(color(green)(1)a + color(purple)(3)b)$

$(a + 3 b) (a + 3 b) = {(a + 3 b)}^{2}$ $(a + 3b)(a + 3b) = (a + 3b)^2$

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How do you factor $a^{2} b + 6 a b^{2} + 9 b^{2}$ $a^2b+6ab^2+9b^2$ ?

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Explanation:

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How do you factor $a^{2} b + 6 a b^{2} + 9 b^{2}$ $a^2b+6ab^2+9b^2$ ?