Find the value of 𝑎 given that two 𝑥 cubed plus 𝑎𝑥 squared minus 21𝑥 minus 36 is divisible by 𝑥 plus four.
Here, we’ve been given a polynomial with a missing coefficient. 𝑎 is the coefficient of 𝑥 squared. And we’re told that this cubic polynomial is divisible by the linear expression 𝑥 plus four. In other words, 𝑥 plus four is a factor of the polynomial two 𝑥 cubed plus 𝑎𝑥 squared minus 21𝑥 minus 36. We can therefore answer this question by applying the factor theorem. This tells us that if 𝑥 plus 𝑎 is a factor of the polynomial 𝑓 of 𝑥, then 𝑓 of negative 𝑎 will be equal to zero, and the converse is also true. If 𝑓 of negative 𝑎 is equal to zero, then we know that 𝑥 plus 𝑎 will be a factor of 𝑓 of 𝑥.
So, we can define 𝑓 of 𝑥 to be our cubic polynomial. And as 𝑥 plus four is a factor of this polynomial, we know that 𝑓 of negative four must be equal to zero. By substituting negative four for 𝑥 and setting the resulting expression equal to zero, we now have an equation which we can solve in order to find the value of 𝑎. Two multiplied by negative four cubed plus 𝑎 multiplied by negative four squared minus 21 multiplied by negative four minus 36 is equal to zero.
By evaluating each of these powers of negative four and then simplifying the equation by grouping like terms, we arrive at the equation 16𝑎 minus 80 is equal to zero. We can then add 80 to each side of the equation and finally divide by 16 to give 𝑎 equals 80 over 16. We can cancel by a factor of eight, first of all, giving 10 over two. And 10 over two is, of course, simply equal to five. So, by using the factor theorem, we’ve found the value of 𝑎, the missing coefficient, in a polynomial 𝑓 of 𝑥. 𝑎 is equal to five.