Answer :
To completely factor the expression [tex]\( 48x^2 + 56x - 8x^3 \)[/tex], let's start by looking for common factors and use the fact that the expression is a polynomial.
1. Identify a common factor:
- The terms in the expression are [tex]\( -8x^3 \)[/tex], [tex]\( 48x^2 \)[/tex], and [tex]\( 56x \)[/tex].
- They all have [tex]\( x \)[/tex] in common, and we can also find that the numerical coefficients [tex]\( -8 \)[/tex], [tex]\( 48 \)[/tex], and [tex]\( 56 \)[/tex] have a greatest common factor of 8.
- Therefore, we can factor out [tex]\(-8x\)[/tex] from all the terms.
2. Factor out the greatest common factor [tex]\( -8x \)[/tex]:
[tex]\[
48x^2 + 56x - 8x^3 = -8x(x^2 - 6x - 7)
\][/tex]
3. Factor the quadratic [tex]\( x^2 - 6x - 7 \)[/tex]:
- We need to factor [tex]\( x^2 - 6x - 7 \)[/tex]. Look for two numbers that multiply to [tex]\(-7\)[/tex] and add up to [tex]\(-6\)[/tex].
- These numbers are [tex]\(-7\)[/tex] and [tex]\(+1\)[/tex].
- So, the quadratic can be factored as [tex]\( (x - 7)(x + 1) \)[/tex].
4. Combine the factors:
- Putting it all together, the completely factored form of the expression is:
[tex]\[
-8x(x - 7)(x + 1)
\][/tex]
Thus, the expression [tex]\( 48x^2 + 56x - 8x^3 \)[/tex] factors completely as [tex]\(-8x(x - 7)(x + 1)\)[/tex].
1. Identify a common factor:
- The terms in the expression are [tex]\( -8x^3 \)[/tex], [tex]\( 48x^2 \)[/tex], and [tex]\( 56x \)[/tex].
- They all have [tex]\( x \)[/tex] in common, and we can also find that the numerical coefficients [tex]\( -8 \)[/tex], [tex]\( 48 \)[/tex], and [tex]\( 56 \)[/tex] have a greatest common factor of 8.
- Therefore, we can factor out [tex]\(-8x\)[/tex] from all the terms.
2. Factor out the greatest common factor [tex]\( -8x \)[/tex]:
[tex]\[
48x^2 + 56x - 8x^3 = -8x(x^2 - 6x - 7)
\][/tex]
3. Factor the quadratic [tex]\( x^2 - 6x - 7 \)[/tex]:
- We need to factor [tex]\( x^2 - 6x - 7 \)[/tex]. Look for two numbers that multiply to [tex]\(-7\)[/tex] and add up to [tex]\(-6\)[/tex].
- These numbers are [tex]\(-7\)[/tex] and [tex]\(+1\)[/tex].
- So, the quadratic can be factored as [tex]\( (x - 7)(x + 1) \)[/tex].
4. Combine the factors:
- Putting it all together, the completely factored form of the expression is:
[tex]\[
-8x(x - 7)(x + 1)
\][/tex]
Thus, the expression [tex]\( 48x^2 + 56x - 8x^3 \)[/tex] factors completely as [tex]\(-8x(x - 7)(x + 1)\)[/tex].
