4. Find the values of $a, b$ and $c$ if $2 x^4+2 x^3+5 x^2+3 x+3=\left(x^2+x+1\right)\left(a x^2+b x+c\right)$.
If $P (x)=2 x^4-3 x^4+m x^2-2 x+24$ has a factor $(2 x+3)$ where $m$ is a constant.
Answer (1)
Expand the right side of the first equation and equate coefficients to find a , b , c .
Substitute x = − 2 3 into the second equation and solve for m .
The values of a , b , c are 2 , 0 , 3 respectively.
The value of m is − 4 39 .
a = 2 , b = 0 , c = 3 , m = − 4 39
Explanation
Problem Analysis We are given two problems. The first problem asks us to find the values of a , b , and c such that 2 x 4 + 2 x 3 + 5 x 2 + 3 x + 3 = ( x 2 + x + 1 ) ( a x 2 + b x + c ) . The second problem states that P ( x ) = 2 x 4 − 3 x 4 + m x 2 − 2 x + 24 has a factor ( 2 x + 3 ) , and we need to find the value of m .
Expanding and Equating Coefficients For the first problem, we expand the right side of the equation: ( x 2 + x + 1 ) ( a x 2 + b x + c ) = a x 4 + b x 3 + c x 2 + a x 3 + b x 2 + c x + a x 2 + b x + c
= a x 4 + ( a + b ) x 3 + ( a + b + c ) x 2 + ( b + c ) x + c
Now, we equate the coefficients of the corresponding terms on both sides of the equation: a = 2
a + b = 2
a + b + c = 5
b + c = 3
c = 3
Solving for a, b, and c From a = 2 and a + b = 2 , we have 2 + b = 2 , so b = 0 . From c = 3 and b + c = 3 , we have b + 3 = 3 , so b = 0 . From a + b + c = 5 , we have 2 + 0 + c = 5 , so c = 3 . Thus, we have a = 2 , b = 0 , c = 3 .
Solving for m For the second problem, we are given P ( x ) = − x 4 + m x 2 − 2 x + 24 and that ( 2 x + 3 ) is a factor of P ( x ) . This means that P ( − 2 3 ) = 0 . Substituting x = − 2 3 into P ( x ) , we get: P ( − 2 3 ) = − ( − 2 3 ) 4 + m ( − 2 3 ) 2 − 2 ( − 2 3 ) + 24 = 0
− 16 81 + m ( 4 9 ) + 3 + 24 = 0
4 9 m = 16 81 − 27 = 16 81 − 432 = − 16 351
m = − 16 351 ⋅ 9 4 = − 4 39
Final Answer Therefore, the values are a = 2 , b = 0 , c = 3 and m = − 4 39 .
Examples
Polynomial factorization and the factor theorem are fundamental concepts in algebra and are used extensively in engineering and computer science. For example, in control systems, the characteristic equation of a system is a polynomial, and the stability of the system can be determined by finding the roots of this polynomial. If we know a factor of the polynomial, we can reduce the complexity of finding the roots. Similarly, in cryptography, polynomial factorization is used in the design of encryption algorithms. Knowing factors helps in simplifying complex systems and solving related equations efficiently.
