To solve for [tex]\( a - b \)[/tex] given the expression [tex]\( 64 - x^3 = (4 - x)(x^2 + a x + b) \)[/tex], we need to equate and compare the coefficients of like terms from both sides of the equation.
1. Expand the right-hand side:
[tex]\[
(4 - x)(x^2 + a x + b) = 4 \cdot x^2 + 4 \cdot a x + 4 \cdot b - x \cdot x^2 - x \cdot a x - x \cdot b
\][/tex]
2. Simplify the expanded expression:
[tex]\[
= 4x^2 + 4ax + 4b - x^3 - ax^2 - bx
\][/tex]
3. Combine like terms:
[tex]\[
= -x^3 + (4 - a)x^2 + (4a - b)x + 4b
\][/tex]
4. Compare it to the left-hand side of the given expression:
[tex]\[
64 - x^3
\][/tex]
5. Match coefficients of corresponding powers of [tex]\( x \)[/tex]:
- The coefficient of [tex]\( x^2 \)[/tex] on the left side is 0 and on the right side is [tex]\( 4 - a \)[/tex], so:
[tex]\[
4 - a = 0
\][/tex]
- The coefficient of [tex]\( x \)[/tex] on the left side is 0 and on the right side is [tex]\( 4a - b \)[/tex], so:
[tex]\[
4a - b = 0
\][/tex]
- The constant term on the left side is 64 and on the right side is 4b, so:
[tex]\[
4b = 64
\][/tex]
6. Solve the equations for [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
- From the equation [tex]\( 4 - a = 0 \)[/tex]:
[tex]\[
a = 4
\][/tex]
- Substituting [tex]\( a = 4 \)[/tex] into [tex]\( 4a - b = 0 \)[/tex]:
[tex]\[
4 \cdot 4 - b = 0 \rightarrow 16 - b = 0 \rightarrow b = 16
\][/tex]
7. Calculate [tex]\( a - b \)[/tex]:
[tex]\[
a - b = 4 - 16 = -12
\][/tex]
Thus, the answer is:
[tex]\[
a - b = -12
\][/tex]
