Answer :
We are given two polynomials:
[tex]$$
P(x) = x^3 - 4x^2 + 3 \quad \text{and} \quad Q(x) = x^3 - 3x^2 - 8x + 19.
$$[/tex]
When a polynomial is divided by [tex]$(x - a)$[/tex], the Remainder Theorem tells us that the remainder is the value of the polynomial at [tex]$x = a$[/tex]. Therefore, the remainders when [tex]$P(x)$[/tex] and [tex]$Q(x)$[/tex] are divided by [tex]$(x-a)$[/tex] are [tex]$P(a)$[/tex] and [tex]$Q(a)$[/tex], respectively.
Since the remainders are equal, we set:
[tex]$$
P(a) = Q(a).
$$[/tex]
Substitute the expressions for [tex]$P(a)$[/tex] and [tex]$Q(a)$[/tex]:
[tex]$$
a^3 - 4a^2 + 3 = a^3 - 3a^2 - 8a + 19.
$$[/tex]
Next, subtract [tex]$a^3$[/tex] from both sides to eliminate the cubic terms:
[tex]$$
-4a^2 + 3 = -3a^2 - 8a + 19.
$$[/tex]
Now, bring all terms to one side to form an equation equal to zero. Add [tex]$4a^2$[/tex] and subtract [tex]$3$[/tex] on both sides:
[tex]$$
0 = (-3a^2 + 4a^2) - 8a + (19 - 3).
$$[/tex]
This simplifies to:
[tex]$$
0 = a^2 - 8a + 16.
$$[/tex]
Notice that the quadratic expression factors perfectly:
[tex]$$
a^2 - 8a + 16 = (a-4)^2.
$$[/tex]
Setting the factor equal to zero:
[tex]$$
(a-4)^2 = 0 \quad \Longrightarrow \quad a-4=0.
$$[/tex]
Thus, we find:
[tex]$$
a = 4.
$$[/tex]
Rounding to the nearest tenth, the value is:
[tex]$$
4.0.
$$[/tex]
Therefore, the final answer is [tex]$\boxed{4.0}$[/tex].
[tex]$$
P(x) = x^3 - 4x^2 + 3 \quad \text{and} \quad Q(x) = x^3 - 3x^2 - 8x + 19.
$$[/tex]
When a polynomial is divided by [tex]$(x - a)$[/tex], the Remainder Theorem tells us that the remainder is the value of the polynomial at [tex]$x = a$[/tex]. Therefore, the remainders when [tex]$P(x)$[/tex] and [tex]$Q(x)$[/tex] are divided by [tex]$(x-a)$[/tex] are [tex]$P(a)$[/tex] and [tex]$Q(a)$[/tex], respectively.
Since the remainders are equal, we set:
[tex]$$
P(a) = Q(a).
$$[/tex]
Substitute the expressions for [tex]$P(a)$[/tex] and [tex]$Q(a)$[/tex]:
[tex]$$
a^3 - 4a^2 + 3 = a^3 - 3a^2 - 8a + 19.
$$[/tex]
Next, subtract [tex]$a^3$[/tex] from both sides to eliminate the cubic terms:
[tex]$$
-4a^2 + 3 = -3a^2 - 8a + 19.
$$[/tex]
Now, bring all terms to one side to form an equation equal to zero. Add [tex]$4a^2$[/tex] and subtract [tex]$3$[/tex] on both sides:
[tex]$$
0 = (-3a^2 + 4a^2) - 8a + (19 - 3).
$$[/tex]
This simplifies to:
[tex]$$
0 = a^2 - 8a + 16.
$$[/tex]
Notice that the quadratic expression factors perfectly:
[tex]$$
a^2 - 8a + 16 = (a-4)^2.
$$[/tex]
Setting the factor equal to zero:
[tex]$$
(a-4)^2 = 0 \quad \Longrightarrow \quad a-4=0.
$$[/tex]
Thus, we find:
[tex]$$
a = 4.
$$[/tex]
Rounding to the nearest tenth, the value is:
[tex]$$
4.0.
$$[/tex]
Therefore, the final answer is [tex]$\boxed{4.0}$[/tex].
