To determine whether [tex]\( (x - 10) \)[/tex] is a factor of the polynomial [tex]\( P(x) = x^3 - 23x^2 + 142x - 120 \)[/tex], we can use the Factor Theorem. The Factor Theorem states that [tex]\( (x - a) \)[/tex] is a factor of the polynomial [tex]\( P(x) \)[/tex] if and only if [tex]\( P(a) = 0 \)[/tex].
### Step-by-Step Solution:
1. Substitute [tex]\( x = 10 \)[/tex] into [tex]\( P(x) \)[/tex]:
[tex]\[
P(10) = (10)^3 - 23(10)^2 + 142(10) - 120
\][/tex]
2. Calculate [tex]\( 10^3 \)[/tex]:
[tex]\[
10^3 = 1000
\][/tex]
3. Calculate [tex]\( 23 \cdot 10^2 \)[/tex]:
[tex]\[
23 \cdot 10^2 = 23 \cdot 100 = 2300
\][/tex]
4. Calculate [tex]\( 142 \cdot 10 \)[/tex]:
[tex]\[
142 \cdot 10 = 1420
\][/tex]
5. Combine all terms:
[tex]\[
P(10) = 1000 - 2300 + 1420 - 120
\][/tex]
6. Perform the arithmetic operations step-by-step:
[tex]\[
1000 - 2300 = -1300
\][/tex]
[tex]\[
-1300 + 1420 = 120
\][/tex]
[tex]\[
120 - 120 = 0
\][/tex]
7. Conclusion:
Since [tex]\( P(10) = 0 \)[/tex], by the Factor Theorem, [tex]\( (x - 10) \)[/tex] is indeed a factor of the polynomial [tex]\( P(x) = x^3 - 23x^2 + 142x - 120 \)[/tex].
### Relation between Assertion (A) and Reason (R):
- Assertion (A): [tex]\( P(x)=x^3 - 23 x^2 + 142 x - 120 \)[/tex], [tex]\( (x-10) \)[/tex] is a factor of the polynomial.
This is true based on the computation above.
- Reason (R): If [tex]\( P(x) \)[/tex] is a polynomial of degree greater than or equal to 1, then [tex]\( (x - a) \)[/tex] is a factor of [tex]\( P(x) \)[/tex] if [tex]\( P(a) = 0 \)[/tex].
This is also true and states the Factor Theorem.
Since both Assertion (A) and Reason (R) are true, and (R) correctly explains (A), we can state that:
### Final Answer:
Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).
