How do you factor $3 x^{2} - 22 x - 16$ $3x^2-22x-16$ ?

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How do you factor $3 x^{2} - 22 x - 16$ $3x^2-22x-16$ ?

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Answer 1 · 2018-04-17T20:46:49

Factor by grouping. $(3 x + 2) (x - 8)$ $(3x+2)(x-8)$

Explanation:

First multiply your $a and b$ $a and b$ terms to get $- 48$ $-48$
Then find two numbers that multiply to that and add up to $- 22$ $-22$
Once you do that rewrite you $b$ $b$ term as those two numbers like so :
$3 x^{2} + 2 x - 24 x - 16$ $3x^2 +2x - 24x -16$
Then put parenthesis around your first and last two terms so you can factor them:
$(3 x^{2} + 2 x) (- 24 x - 16)$ $(3x^2 +2x) (- 24x -16)$
Then factor normally:
$x (3 x + 2) - 8 (3 x + 2)$ $x(3x+2) -8(3x+2)$
your two parenthesis should be the same and if they are cross out one of them and make a new parenthesis with the two numbers you factored out:
$(3 x + 2) (x - 8)$ $(3x+2)(x-8)$

Answer 2 · 2018-04-17T20:53:38

(3x+2)(x-8)

Explanation:

We begin with $3 x^{2} - 22 x - 16$ $3x^2-22x-16$ Automatically we can set up this..
$(3 x + ?) (x + ?)$ $(3x+?)(x+?)$ Since $3 x^{2}$ $3x^2$ could only factor into $3 x$ $3x$ and $x$ $x$
Now to fill in the ?'s
The question marks must multiply to equal 16 and add up to -22.
You can always use the guess and check method until you come across the correct answer or set up some equations..
$3 x + y = 22$ $3x+y=22$
$x y = - 16$ $xy=-16$
Then to solve this system of equations you can use subsitution...

Answer 3 · 2018-04-17T21:28:29

$(x - 8) (3 x + 2)$ $(x-8)(3x+2)$

Explanation:

These are not always easy to spot. Sometimes you really have to 'play' with them to find the correct combination.

First lets look at the constant of $- 16$ $-16$

As this is negative it means that it is the product of a negative and a positive value.

What are all the whole number factors of 16?

$1 \times 16; 2 \times 8; 4 \times 4$ $1xx16; 2xx8; 4xx4$

3 is a prime number so we can only have $1 \times 3$ $1xx3$ giving:

$(x \pm ?) (3 x \pm ?)$ $(x+-?)(3x+-?)$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
I am deliberately 'going round the houses' to make a point!

$Try 1 and 16 for - 22 x$ $color(brown)("Try 1 and 16 for "-22x)$

$(x \pm ?) (3 x \pm ?) \to (x \pm 1) (3 x \pm 16)$ $(x+-?)(3x+-?) -> (x+-1)(3x+-16)$

$\pm 16 \pm 3 \neq - 22 : dd \pm (3 \times 16) \pm 1 = - 22 \leftarrow Fail$ $+-16+-3!=-22: color(white)("dd") +-(3xx16)+-1=-22 color(red)(larr"Fail")$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$Try 4 and 4 for - 22 x$ $color(brown)("Try 4 and 4 for "-22x)$

$(x \pm ?) (3 x \pm ?) \to (x \pm 4) (3 x \pm 4)$ $(x+-?)(3x+-?) -> (x+-4)(3x+-4)$

$\pm (3 \times 4) \pm 4 \neq - 22 \leftarrow Fail$ $+-(3xx4)+-4!=-22 color(red)(larr" Fail")$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$Try 2 and 8 for - 22 x$ $color(brown)("Try 2 and 8 for "-22x)$

$(x \pm ?) (3 x \pm ?) \to (x \pm 2) (3 x \pm 8)$ $(x+-?)(3x+-?) -> (x+-2)(3x+-8)$

$(3 \times 2) - 8 \neq - 22 \leftarrow Fail$ $(3xx2)-8!=-22 color(red)(larr" Fail")$

$(3 \times (- 8)) + 2 = - 22 \leftarrow WORKS!$ $(3xx(-8))+2 =-22 color(green)(larr" WORKS!")$

$(x - 8) (3 x + 2)$ $(x-8)(3x+2)$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$Check$ $color(blue)("Check")$

$+ x (3 x + 2) = 3 x^{2} + 2 x$ $+x(3x+2) = 3x^2+2x$
$-8(3x+2)=ul(color(white)(ccd.) -24x-16)$
$color(white)("dddddddddddd")3x^2-22x-16 larr" As required"$

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How do you factor $3 x^{2} - 22 x - 16$ $3x^2-22x-16$ ?

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