Answer :
To verify that [tex]$-4$[/tex] is a solution to the equation
[tex]$$6x^3 + 19x^2 - 19x + 4 = 0,$$[/tex]
we substitute [tex]$x = -4$[/tex] into the polynomial and compute the result step by step.
1. Evaluate the first term:
[tex]$$6(-4)^3 = 6(-64) = -384.$$[/tex]
2. Evaluate the second term:
[tex]$$19(-4)^2 = 19(16) = 304.$$[/tex]
3. Evaluate the third term:
[tex]$$-19(-4) = 76.$$[/tex]
4. The constant term remains:
[tex]$$4.$$[/tex]
Now, add all the terms together:
[tex]$$-384 + 304 + 76 + 4.$$[/tex]
5. Compute the sum:
[tex]\[
-384 + 304 = -80,
\][/tex]
[tex]\[
-80 + 76 = -4,
\][/tex]
[tex]\[
-4 + 4 = 0.
\][/tex]
Since the sum is [tex]$0$[/tex], the remainder when [tex]$6x^3 + 19x^2 - 19x + 4$[/tex] is divided by [tex]$(x + 4)$[/tex] is zero. This confirms that [tex]$-4$[/tex] is indeed a solution to the equation.
[tex]$$6x^3 + 19x^2 - 19x + 4 = 0,$$[/tex]
we substitute [tex]$x = -4$[/tex] into the polynomial and compute the result step by step.
1. Evaluate the first term:
[tex]$$6(-4)^3 = 6(-64) = -384.$$[/tex]
2. Evaluate the second term:
[tex]$$19(-4)^2 = 19(16) = 304.$$[/tex]
3. Evaluate the third term:
[tex]$$-19(-4) = 76.$$[/tex]
4. The constant term remains:
[tex]$$4.$$[/tex]
Now, add all the terms together:
[tex]$$-384 + 304 + 76 + 4.$$[/tex]
5. Compute the sum:
[tex]\[
-384 + 304 = -80,
\][/tex]
[tex]\[
-80 + 76 = -4,
\][/tex]
[tex]\[
-4 + 4 = 0.
\][/tex]
Since the sum is [tex]$0$[/tex], the remainder when [tex]$6x^3 + 19x^2 - 19x + 4$[/tex] is divided by [tex]$(x + 4)$[/tex] is zero. This confirms that [tex]$-4$[/tex] is indeed a solution to the equation.
