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Consider this polynomial, where $a$ is an unknown real number:

$p(x)=x^4+5 x^3+a x^2-3 x+11$



The remainder of the quotient of $p(x)$ and $(x+1)$ 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 [Choose an answer] found the value of $a$ because he [Choose an answer].



Zahra [Choose an answer] found the value of $a$ because she [Choose an answer].

Verified Answer :

  • Both the Remainder Theorem and synthetic division confirm that the remainder of \(p(x) = x^4 + 5x^3 + ax^2 - 3x + 11\) divided by \((x + 1)\) is \(10 + a\).
  • Setting \(10 + a = 17\) confirms the value \(a = 7\).
  • Therefore, both Braulio and Zahra reached the correct result through their respective valid mathematical methods.
  • The final value is \(\boxed{a = 7}\).

Explanation

  1. Analysis using the Remainder Theorem

To find the value of \(a\) for the polynomial \(p(x) = x^4 + 5x^3 + ax^2 - 3x + 11\), we use two different methods: the Remainder Theorem and synthetic division. The Remainder Theorem states that the remainder of the division of a polynomial \(p(x)\) by \((x - c)\) is \(p(c)\). For division by \((x + 1)\), we have \(c = -1\). Calculating \(p(-1)\), we get:
\[p(-1) = (-1)^4 + 5(-1)^3 + a(-1)^2 - 3(-1) + 11 = 1 - 5 + a + 3 + 11 = 10 + a.\]
Since the remainder is given as 17, we set \(10 + a = 17\), which gives \(a = 7\).

  1. Analysis using Synthetic Division

Synthetic division is a shorthand method for dividing a polynomial by a linear factor of the form \((x - c)\). To divide \(p(x) = x^4 + 5x^3 + ax^2 - 3x + 11\) by \((x + 1)\), we set up the synthetic division with coefficients \([1, 5, a, -3, 11]\) and evaluate at \(c = -1\):

  1. Bring down the leading coefficient: \(1\).
  2. Multiply by \(-1\): \(1 \times -1 = -1\). Add to the next coefficient: \(5 + (-1) = 4\).
  3. Multiply by \(-1\): \(4 \times -1 = -4\). Add to the next coefficient: \(a + (-4) = a - 4\).
  4. Multiply by \(-1\): \((a - 4) \times -1 = -a + 4\). Add to the next coefficient: \(-3 + (-a + 4) = 1 - a\).
  5. Multiply by \(-1\): \((1 - a) \times -1 = a - 1\). Add to the next coefficient: \(11 + (a - 1) = 10 + a\).

The remainder from the synthetic division is \(10 + a\). Setting this equal to 17, we find \(a = 7\). Thus, both methods are mathematically consistent and yield the correct result.

  1. Conclusion

Since both Braulio and Zahra correctly applied their respective methods (synthetic division and the remainder theorem), they both correctly identified the value of \(a\).
Braulio \(\boxed{\text{correctly}}\) found the value of \(a\) because he \(\boxed{\text{correctly set up and performed synthetic division}}\).
Zahra \(\boxed{\text{correctly}}\) found the value of \(a\) because she \(\boxed{\text{correctly applied the remainder theorem}}\).

Examples

In fields like computer science and data analysis, checking remainders of polynomial divisions is used in error detection and correction codes. For example, cyclic redundancy checks (CRC) use polynomial division to ensure data integrity during transmission. Understanding that synthetic division and the remainder theorem are equivalent allows engineers to choose the most efficient method for implementing these checks.

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