When a polynomial is divided by [tex]$4x-6$[/tex], the quotient is [tex]$2x^2+x+1$[/tex] and the remainder is -4. What is the dividend, [tex]$f(x)$[/tex]? Explain.
Answer (2)
By using the polynomial division algorithm, we substitute the given divisor, quotient, and remainder to find the polynomial f ( x ) . After expanding and simplifying, we find that the dividend is 8 x 3 − 8 x 2 − 2 x − 10 .
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∙ Use the division algorithm for polynomials: Dividend = ( Divisor ) × ( Quotient ) + Remainder .
∙ Substitute the given divisor, quotient, and remainder into the equation: f ( x ) = ( 4 x − 6 ) ( 2 x 2 + x + 1 ) + ( − 4 ) .
∙ Expand and simplify the expression: f ( x ) = 8 x 3 − 8 x 2 − 2 x − 6 − 4 .
∙ The dividend is: f ( x ) = 8 x 3 − 8 x 2 − 2 x − 10 .
Explanation
Understanding the Problem We are given that when a polynomial f ( x ) is divided by 4 x − 6 , the quotient is 2 x 2 + x + 1 and the remainder is − 4 . We want to find the dividend, which is the polynomial f ( x ) .
Applying the Division Algorithm We can use the division algorithm for polynomials, which states that Dividend = ( Divisor ) × ( Quotient ) + Remainder .
Substituting the Given Values In this case, the divisor is 4 x − 6 , the quotient is 2 x 2 + x + 1 , and the remainder is − 4 . Substituting these values into the division algorithm, we get f ( x ) = ( 4 x − 6 ) ( 2 x 2 + x + 1 ) + ( − 4 ) .
Expanding the Expression Now, we need to expand the expression ( 4 x − 6 ) ( 2 x 2 + x + 1 ) . We can do this by multiplying each term in the first factor by each term in the second factor:
( 4 x − 6 ) ( 2 x 2 + x + 1 ) = 4 x ( 2 x 2 + x + 1 ) − 6 ( 2 x 2 + x + 1 ) = 8 x 3 + 4 x 2 + 4 x − 12 x 2 − 6 x − 6 = 8 x 3 + ( 4 x 2 − 12 x 2 ) + ( 4 x − 6 x ) − 6 = 8 x 3 − 8 x 2 − 2 x − 6.
Adding the Remainder Now, we add the remainder − 4 to the expanded expression:
f ( x ) = ( 8 x 3 − 8 x 2 − 2 x − 6 ) + ( − 4 ) f ( x ) = 8 x 3 − 8 x 2 − 2 x − 6 − 4 f ( x ) = 8 x 3 − 8 x 2 − 2 x − 10.
Final Answer Therefore, the dividend, f ( x ) , is 8 x 3 − 8 x 2 − 2 x − 10 .
Examples
Polynomial division is used in various applications, such as finding the roots of a polynomial, simplifying complex expressions, and designing control systems. For example, in engineering, polynomial division can help determine the stability of a system by analyzing the roots of its characteristic equation. In computer graphics, it can be used to model curves and surfaces. Understanding polynomial division provides a foundation for solving real-world problems in science and engineering.
