Question - Finding Coefficients of a Polynomial
Solution:
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 $$ b = -22.5 $$.
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