Finding Coefficients of a Polynomial
The problem provides a polynomial \( p(x) = 5x^3 + 9x^2 + ax + b \) and states two conditions:
1. \( x - 2 \) is a factor of \( p(x) \).
2. The remainder is 27 when \( p(x) \) is divided by \( x + 2 \).
Let's use these conditions to find the values of a and b.
For the first condition, since \( x - 2 \) is a factor of \( p(x) \), \( p(2) = 0 \). So, we substitute x = 2 into the polynomial:
\( p(2) = 5(2)^3 + 9(2)^2 + a(2) + b = 0 \)
\( 40 + 36 + 2a + b = 0 \)
\( 76 + 2a + b = 0 \).
From this equation, we will extract the relationship between a and b.
Now let's address the second condition. The remainder when \( p(x) \) is divided by \( x + 2 \) is found by evaluating \( p(-2) \).
\( p(-2) = 5(-2)^3 + 9(-2)^2 - 2a + b \)
\( -40 + 36 - 2a + b = 27 \)
\( -4 - 2a + b = 27 \)
\( -2a + b = 31 \).
Now we have two equations with two variables:
1. \( 76 + 2a + b = 0 \)
2. \( -2a + b = 31 \).
We can solve this system of equations by adding or subtracting them. Let's subtract the second equation from the first:
\( (76 + 2a + b) - (-2a + b) = 0 - 31 \)
\( 76 + 2a + b + 2a - b = -31 \)
\( 76 + 4a = -31 \)
\( 4a = -31 - 76 \)
\( 4a = -107 \)
\( a = -107 / 4 \)
\( a = -26.75 \).
Now, let's substitute \( a = -26.75 \) into one of the equations to find b. We can use the second equation for easier calculation:
\( -2(-26.75) + b = 31 \)
\( 53.5 + b = 31 \)
\( b = 31 - 53.5 \)
\( b = -22.5 \).
Thus, the values are \( a = -26.75 \) and \( b = -22.5 \).
