Question

1. when $ax^{3}-x^{2}+2x + b$ is divided by $x - 1$, the remainder is 10. when it is divided by $x - 2$, the remainder is 51. find the values of $a$ and $b$.

Explanation:

Step1: Apply Remainder Theorem

Let \(f(x)=ax^{3}-x^{2}+2x + b\). When \(f(x)\) is divided by \(x - 1\), \(f(1)=a\times1^{3}-1^{2}+2\times1 + b=10\), so \(a + b+1 = 10\), or \(a + b=9\).

Step2: Apply Remainder Theorem again

When \(f(x)\) is divided by \(x - 2\), \(f(2)=a\times2^{3}-2^{2}+2\times2 + b=51\), so \(8a + b=51\).

Step3: Solve the system of equations

Subtract the first - equation from the second: \((8a + b)-(a + b)=51 - 9\), \(7a=42\), \(a = 6\). Substitute \(a = 6\) into \(a + b=9\), we get \(6 + b=9\), \(b = 3\).

Answer:

\(a = 6\), \(b = 3\)