Question

What is the remainder when \(f(x)=-3 x^{17}+x^{9}-x^{5}+2 x\) is divided by \(x+1 ?\)

Short answer

The remainder is 1.

Step-by-step solution

Identify the divisor

The divisor given is \(x + 1\).

Use the Remainder Theorem

According to the Remainder Theorem, if a polynomial \(f(x)\) is divided by \(x - c\), the remainder is \(f(c)\). Here, the divisor is written as \(x + 1\), which can be rewritten as \(x - (-1)\). Therefore, \(c = -1\).

Substitute \(c = -1\) into \(f(x)\)

Substitute \((-1)\) into the polynomial \(f(x)\): \[ f(-1) = -3(-1)^{17} + (-1)^{9} - (-1)^{5} + 2(-1) \] Simplify each term: \[ -3(-1)^{17} = -3(-1) = 3 \] \[ (-1)^{9} = -1 \] \[ -(-1)^{5} = -(-1) = 1 \] \[ 2(-1) = -2 \] Combining these results: \[ f(-1) = 3 - 1 + 1 - 2 \]

Simplify the expression

Simplify the expression to find the remainder: \[ f(-1) = 3 - 1 + 1 - 2 \] \[ f(-1) = 1 \]

Key concepts

Polynomial Division

Polynomial division is a method used to divide one polynomial by another. It is similar to the long division of numbers. To perform polynomial division, you need to know the degree of each polynomial involved. The degree is the highest power of the variable in the polynomial.
In our example, we have a polynomial \(f(x) = -3x^{17} + x^9 - x^5 + 2x\) and a divisor \(x + 1\). The goal is to find the remainder when \(f(x)\) is divided by \(x + 1\). Practicing polynomial division can be a bit challenging at first, but understanding the steps can make it much easier. This technique is essential for solving many algebraic problems and can simplify complex expressions substantially.

Remainder

The remainder is what is left over after performing a division operation. In polynomial division, the remainder is the value left after dividing the polynomial by the divisor. When you use the Remainder Theorem, finding the remainder becomes simpler. The Remainder Theorem states that for a polynomial \(f(x)\), if you divide by \(x - c\), then the remainder is \(f(c)\).
In our exercise, the divisor is \(x + 1\). By transforming it to \(x - (-1)\), we find that \(c = -1\). By evaluating \(f(x)\) at \(c\), we directly get the remainder without performing long division. Plugging in \(-1\) into \(f(x)\) simplifies the process and shows the power of the theorem.

Substitution Method

The substitution method is a technique where you replace the variable with a specific value to simplify an expression. This method is particularly useful when using the Remainder Theorem for polynomials.
In the given problem, we substitute \(-1\) for \(x\) in the polynomial \(f(x)\). This is because our divisor \(x + 1\) translates to \(x - (-1)\), meaning we need to evaluate \(f(-1)\). Substituting \(-1\) into \(-3(-1)^{17} + (-1)^{9} - (-1)^{5} + 2(-1)\), we perform the calculations step by step:

• \(-3(-1)^{17}\) simplifies to \(3\)
• \((-1)^9\) simplifies to \(-1\)
• \(-(-1)^5\) simplifies to \(1\)
• \(2(-1)\) simplifies to \(-2\)

Combining these gives us the remainder: \(3 - 1 + 1 - 2 = 1\). This substitution method is straightforward and ensures you get the correct remainder efficiently.