Consider this polynomial, where $a$ is an unknown real number.
[tex]p(x)=x^4+5 x^3+a x^2-3 x+11[/tex]
The remainder of the quotient of [tex]p(x)[/tex], and [tex](x+1)[/tex] is 17.
Braulio uses synthetic division to find the value of $a$, and Zahra uses the remainder theorem to find the value of $a$. Their work is shown.
Braulio found the value of a because he
Zahra found the value of a because she
Answer (2)
Apply the Remainder Theorem: p ( − 1 ) = 17 .
Substitute x = − 1 into p ( x ) : ( − 1 ) 4 + 5 ( − 1 ) 3 + a ( − 1 ) 2 − 3 ( − 1 ) + 11 = 17 .
Simplify the equation: 1 − 5 + a + 3 + 11 = 17 .
Solve for a : a = 7 , so the final answer is 7 .
Explanation
Understanding the Problem We are given the polynomial p ( x ) = x 4 + 5 x 3 + a x 2 − 3 x + 11 , where a is an unknown real number. We also know that when p ( x ) is divided by ( x + 1 ) , the remainder is 17. Braulio uses synthetic division, and Zahra uses the remainder theorem to find the value of a . We need to determine if they found the correct value of a and explain why.
Applying the Remainder Theorem The Remainder Theorem states that if we divide a polynomial p ( x ) by ( x − c ) , the remainder is p ( c ) . In this case, we are dividing by ( x + 1 ) , so c = − 1 . Therefore, p ( − 1 ) = 17 .
Substitution Substitute x = − 1 into the polynomial p ( x ) :
p ( − 1 ) = ( − 1 ) 4 + 5 ( − 1 ) 3 + a ( − 1 ) 2 − 3 ( − 1 ) + 11
Simplifying Simplify the expression: p ( − 1 ) = 1 − 5 + a + 3 + 11 = 17
Combining Constants Combine the constants: 1 − 5 + 3 + 11 + a = 10 + a = 17
Solving for a Solve for a :
a = 17 − 10 = 7
Conclusion Since Zahra used the Remainder Theorem correctly, she found the correct value of a , which is 7. Braulio used synthetic division, which is an equivalent method to the Remainder Theorem. If he performed the synthetic division correctly, he should also find a = 7 .
Examples
Polynomials are used in many areas of mathematics and science. For example, they are used to model curves and trajectories in physics, such as the path of a projectile. In economics, polynomials can be used to represent cost and revenue functions, helping businesses analyze their profitability. In computer graphics, polynomials are used to create smooth curves and surfaces for 3D models.
Braulio successfully found the value of a using synthetic division, while Zahra employed the Remainder Theorem to reach the same conclusion. Both methods lead to the correct answer of a = 7 . Therefore, both students are correct in their approaches.
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