Answer :
We begin with the expression
[tex]$$45x^4 - 60x^3.$$[/tex]
Step 1. Factor out the greatest common factor (GCF):
Both terms share a factor of [tex]$x^3$[/tex]. Factoring [tex]$x^3$[/tex] out, we have
[tex]$$45x^4 - 60x^3 = x^3(45x - 60).$$[/tex]
Step 2. Factor the binomial:
Now, look at the binomial inside the parentheses: [tex]$45x - 60$[/tex]. Both terms in this binomial have a common numerical factor of [tex]$15$[/tex]. Factoring out [tex]$15$[/tex], we obtain
[tex]$$45x - 60 = 15(3x - 4).$$[/tex]
Step 3. Combine the factors:
Substitute the factored binomial back into the expression:
[tex]$$45x^4 - 60x^3 = x^3 \cdot 15(3x - 4) = 15x^3(3x - 4).$$[/tex]
Thus, the expression [tex]$45x^4 - 60x^3$[/tex] is equivalent to
[tex]$$15x^3(3x-4),$$[/tex]
which shows that the coefficient [tex]$r$[/tex] in [tex]$r x^3 (3x - 4)$[/tex] is
[tex]$$r = 15.$$[/tex]
Summary of intermediate factors:
- The greatest common factor of the original expression is [tex]$x^3$[/tex].
- The common numerical factor in the binomial is [tex]$15$[/tex].
- The resulting binomial after factoring is [tex]$3x-4$[/tex] with coefficients [tex]$3$[/tex] and [tex]$-4$[/tex].
This completes the detailed step-by-step solution.
[tex]$$45x^4 - 60x^3.$$[/tex]
Step 1. Factor out the greatest common factor (GCF):
Both terms share a factor of [tex]$x^3$[/tex]. Factoring [tex]$x^3$[/tex] out, we have
[tex]$$45x^4 - 60x^3 = x^3(45x - 60).$$[/tex]
Step 2. Factor the binomial:
Now, look at the binomial inside the parentheses: [tex]$45x - 60$[/tex]. Both terms in this binomial have a common numerical factor of [tex]$15$[/tex]. Factoring out [tex]$15$[/tex], we obtain
[tex]$$45x - 60 = 15(3x - 4).$$[/tex]
Step 3. Combine the factors:
Substitute the factored binomial back into the expression:
[tex]$$45x^4 - 60x^3 = x^3 \cdot 15(3x - 4) = 15x^3(3x - 4).$$[/tex]
Thus, the expression [tex]$45x^4 - 60x^3$[/tex] is equivalent to
[tex]$$15x^3(3x-4),$$[/tex]
which shows that the coefficient [tex]$r$[/tex] in [tex]$r x^3 (3x - 4)$[/tex] is
[tex]$$r = 15.$$[/tex]
Summary of intermediate factors:
- The greatest common factor of the original expression is [tex]$x^3$[/tex].
- The common numerical factor in the binomial is [tex]$15$[/tex].
- The resulting binomial after factoring is [tex]$3x-4$[/tex] with coefficients [tex]$3$[/tex] and [tex]$-4$[/tex].
This completes the detailed step-by-step solution.
