Solve this. Factor out the GCF of the polynomial. `56x^7+ 21x^4+ 63x^3`
`56x^7+ 21x^4+ 63x^3`
First, determine the GCF. To do so, express each term with their prime factors.
`56x^7 = 7*8*x*x*x*x*x*x*x`
`21x^4 = 3*7*x*x*x*x`
Then, look for the factors that is present in all terms.
56x^7 = 7`*` 8`*` x`*` x`*` x`*` x`*` x`*` x`*` x
21x^4 = 3`*` 7`*` x`*` x`*` x`*` x
63x^3 = 7*9*x*x*x
Hence, the GCF is `7x^3` .
Now that the GCF is known, factor it out.
So, its factor form is:
`56x^7+ 21x^4+ 63x^3`
`=7x^3( __ + __ + __ )`
To determine what goes inside the parenthesis, divide each term by the GCF.
For the first blank, `(56x^7)/(7x^3)=8x^(7-3)=8x^4` .
For the second blank, `(21x^4)/(7x^3)=3x^(4-3)=3x^1=3` x.
And, for the third blank,` (63x^3)/(7x^3)=9x^(3-3)=9x^0=` 9.
Therefore, `56x^7+ 21x^4+ 63x^3=7x^3(8x^4+3x+9)` .
Factor GCF of ` 56x^7+21x^4+63x^3 ` .
GCF is the largest factor that can divide all of the terms evenly.
56, 21,and 63 has a GCF of: 7
`x^7, x^4, x^3 ` all have a GCF of `x^3 ` .
Therefore the GCF is: `7x^3 ` .
Divide each term by the GCF of `7x^3`
`(56x^7)/(7x^3) + (21x^4)/(7x^3) + (63x^3)/(7x^3) `
`8x^4 + 3x + 9 ` `lArr` Final Answer
`56x^7+21x^4+63x^3 ` find the greatest common factor which would be `7x^3 ` factor it out
`7x^3(8x^4+3x+9)` and that's the answer
56x^7= 7*8*x*x*x*x*x*x*x Greatest Common Factor: 7x^3
The Greatest Common Factor is the highest number which divides the polynomial.
We use the following property to solve the question:
`x^(a+b) = x^a xx x^b`
Now we will break the polynomial to get the common factor
Given equation: `56x^7 + 21x^4 + 63x^3`
`= 7xx8xxx^(3+4) + 7xx3xxx^(3+1) + 7xx9xxx^3`
`= 7 xx 8 xx x^3 xx x^4 + 7xx3xxx^3 xx x^1 + 7 xx 9xx x^3`
In order to factor out a GCF from the expression 56x^7+21x^4+63x^3, there must be something that all three terms have in common.
Look first at the variables. What do they all have in common?
All have at least three x's within the term. Therefore, we can take an x^3 out from each term, and the result is: x^3(56x^4+21x^1+63).
Then look at the coefficents. What factors do the coefficients have in common?
When you factor 56, you have 7 x 2 x 2 x 2. Factor 21, and you get 7 x 3. Likewise, 63 factors into 7 x 3 x 3 x 3. Clearly, all of the coefficients have a 7 in common, so that is also factored out of the expression and joins the factored x^3. The result:
7x^3(8x^4+3x^1+9). The term 7x^3 is the greatest common factor of all three terms in the original equation.