Math / Algebra

QuestionFactor the polynomial by grouping. Do not use extra spaces in your answers. $4 x^{2}-16 x+3 x-12$

Studdy Solution

STEP 1

What is this asking? We need to rewrite this polynomial as a product of two smaller polynomials. Watch out! Don't forget to factor completely!
Sometimes, after an initial grouping, there might be more factoring that can be done.

STEP 2

1. Group the Terms
2. Factor Each Group
3. Factor Out the Common Binomial

STEP 3

Alright, let's group the first two terms and the last two terms together.
This sets us up perfectly to find common factors within each group.
So we have $(4x^2 - 16x) + (3x - 12)$ .
Remember, we're strategically grouping these terms to make our factoring easier!

STEP 4

Now, let's factor out the greatest common factor (GCF) from each group.
In the first group, $(4x^2 - 16x)$ , the GCF is $4x$ .
Factoring that out gives us $4x(x - 4)$ .
See how that nicely isolates the $(x-4)$ ?

STEP 5

For the second group, $(3x - 12)$ , the GCF is 3.
Factoring that out gives us $3(x - 4)$ .
Boom! Notice how we now have a common binomial factor, $(x - 4)$ , in both groups.
This is key!

STEP 6

Since both groups share the common factor $(x - 4)$ , we can factor it out.
Think of $(x - 4)$ as a single term that's being multiplied by both $4x$ and $3$ .
So, we can rewrite the entire expression as $(x - 4)(4x + 3)$ .
We're essentially using the distributive property in reverse!

STEP 7

So, our fully factored polynomial is $(x - 4)(4x + 3)$ .
We've successfully rewritten our original polynomial as a product of two simpler ones!

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QuestionFactor the polynomial by grouping. Do not use extra spaces in your answers. $4 x^{2}-16 x+3 x-12$

Studdy Solution

STEP 1

What is this asking? We need to rewrite this polynomial as a product of two smaller polynomials. Watch out! Don't forget to factor completely!
Sometimes, after an initial grouping, there might be more factoring that can be done.

STEP 2

1. Group the Terms
2. Factor Each Group
3. Factor Out the Common Binomial

STEP 3

Alright, let's group the first two terms and the last two terms together.
This sets us up perfectly to find common factors within each group.
So we have $(4x^2 - 16x) + (3x - 12)$ .
Remember, we're strategically grouping these terms to make our factoring easier!

STEP 4

Now, let's factor out the greatest common factor (GCF) from each group.
In the first group, $(4x^2 - 16x)$ , the GCF is $4x$ .
Factoring that out gives us $4x(x - 4)$ .
See how that nicely isolates the $(x-4)$ ?

STEP 5

For the second group, $(3x - 12)$ , the GCF is 3.
Factoring that out gives us $3(x - 4)$ .
Boom! Notice how we now have a common binomial factor, $(x - 4)$ , in both groups.
This is key!

STEP 6

Since both groups share the common factor $(x - 4)$ , we can factor it out.
Think of $(x - 4)$ as a single term that's being multiplied by both $4x$ and $3$ .
So, we can rewrite the entire expression as $(x - 4)(4x + 3)$ .
We're essentially using the distributive property in reverse!

STEP 7

So, our fully factored polynomial is $(x - 4)(4x + 3)$ .
We've successfully rewritten our original polynomial as a product of two simpler ones!

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QuestionFactor the polynomial by grouping. Do not use extra spaces in your answers. 4x2−16x+3x−124 x^{2}-16 x+3 x-124x2−16x+3x−12

Studdy Solution

STEP 1

What is this asking? We need to rewrite this polynomial as a product of two smaller polynomials. Watch out! Don't forget to factor completely!Sometimes, after an initial grouping, there might be more factoring that can be done.

STEP 2

1. Group the Terms2. Factor Each Group3. Factor Out the Common Binomial

STEP 3

STEP 4

Now, let's **factor out** the **greatest common factor (GCF)** from each group.In the first group, (4x2−16x)(4x^2 - 16x)(4x2−16x), the GCF is 4x4x4x.Factoring that out gives us 4x(x−4)4x(x - 4)4x(x−4).See how that nicely isolates the (x−4)(x-4)(x−4)?

STEP 5

For the second group, (3x−12)(3x - 12)(3x−12), the GCF is **3**.Factoring that out gives us 3(x−4)3(x - 4)3(x−4).Boom! Notice how we now have a common binomial factor, (x−4)(x - 4)(x−4), in both groups.This is key!

STEP 6

STEP 7

So, our fully factored polynomial is (x−4)(4x+3)(x - 4)(4x + 3)(x−4)(4x+3).We've successfully rewritten our original polynomial as a product of two simpler ones!

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionFactor the polynomial by grouping. Do not use extra spaces in your answers. $4 x^{2}-16 x+3 x-12$

What is this asking? We need to rewrite this polynomial as a product of two smaller polynomials. Watch out! Don't forget to factor completely!
Sometimes, after an initial grouping, there might be more factoring that can be done.

1. Group the Terms
2. Factor Each Group
3. Factor Out the Common Binomial

Now, let's factor out the greatest common factor (GCF) from each group.
In the first group, $(4x^2 - 16x)$ , the GCF is $4x$ .
Factoring that out gives us $4x(x - 4)$ .
See how that nicely isolates the $(x-4)$ ?

For the second group, $(3x - 12)$ , the GCF is 3.
Factoring that out gives us $3(x - 4)$ .
Boom! Notice how we now have a common binomial factor, $(x - 4)$ , in both groups.
This is key!

So, our fully factored polynomial is $(x - 4)(4x + 3)$ .
We've successfully rewritten our original polynomial as a product of two simpler ones!