Question

Find the remainder when \(P(x)=8 x^{4}-2 x^{3}+x-8\) is divided hy \(x+2\)

Short answer

The remainder is 134.

Step-by-step solution

- Identify the divisor and use the Remainder Theorem

The divisor is given as \(x+2\). According to the Remainder Theorem, the remainder of a polynomial \(P(x)\) when divided by \(x-c\) is \(P(c)\). Here, we need \(c\) such that \(x+2 = 0\). Solving this, we get \(c = -2\).

- Evaluate the polynomial at \x = -2\

Substitute \(x = -2\) into \(P(x) = 8x^4 - 2x^3 + x - 8\): \[P(-2) = 8(-2)^4 - 2(-2)^3 + (-2) - 8\].

- Simplify the expression

Calculate each term step-by-step: \[8(-2)^4 = 8 \times 16 = 128\], \[-2(-2)^3 = -2 \times (-8) = 16\], \[-2 = -2\], and \[-8 = -8\]. Combine all these values: \[128 + 16 - 2 - 8 = 134\].

- State the remainder

The remainder when \(P(x) = 8x^4 - 2x^3 + x - 8\) is divided by \(x+2\) is \[134\].

Key concepts

Polynomial Division

Understanding polynomial division is crucial when working with polynomials. Polynomial division involves dividing one polynomial by another, resulting in a quotient and a remainder. This concept is similar to long division but applied to polynomials.
It's common in algebra and calculus, and it helps to simplify complex polynomial expressions. When dividing, ensure that the polynomials are arranged in descending order of their exponents for easier manipulation.
For example, if you divide the polynomial by another polynomial of a lower degree, you will get a quotient polynomial and possibly a remainder. Specifically, for a polynomial division: \(P(x) = D(x) \times Q(x) + R(x)\)
- \(P(x)\) is the polynomial being divided.
- \(D(x)\) is the divisor.
- \(Q(x)\) is the quotient.
- \(R(x)\) is the remainder, and its degree is less than the degree of the divisor.

Evaluating Polynomials

Evaluating polynomials means finding the value of a polynomial function at a given point. This is often done by substituting a specific value for \(x\) in the polynomial expression.
You can follow these steps to evaluate a polynomial:

• Identify the polynomial and the value at which you want to evaluate it.
• Substitute the given value into the polynomial.
• Simplify the polynomial expression to find the result.

For instance, if given the polynomial \(P(x) = 2x^3 - 3x^2 + x - 5\), to evaluate it at \(x = 2\), substitute 2 for \(x\): \(P(2) = 2(2)^3 - 3(2)^2 + 2 - 5\).
Then calculate each term's value and sum them up. This process helps determine the polynomial's behavior at specific points, which is useful in graphing and analyzing polynomial functions.

Finding the Remainder

Finding the remainder when dividing polynomials is simplified using the Remainder Theorem. This theorem states that for a polynomial \(P(x)\) divided by \(x-c\), the remainder is simply \(P(c)\).
This method bypasses the need for long division and provides a quick way to find the remainder.
Here's the step-by-step approach using the provided example:

• Identify the divisor: Here, it is \(x+2\).
• Set the divisor equal to 0 to find \(c\): \(x+2=0 \rightarrow x=-2\), so \(c=-2\).
• Evaluate the polynomial at \(x=-2\): \(P(x)=8x^4 -2x^3 + x -8\).
  Substitute \(x=-2\): \(P(-2) = 8(-2)^4 - 2(-2)^3 + (-2) - 8\).
• Calculate each term: \(8(-2)^4 = 128\), \(-2(-2)^3 = 16\), \(-2 = -2\), \(-8 = -8\).
• Add the values: \(128 + 16 - 2 - 8 = 134\).

Therefore, the remainder when \(P(x) = 8x^4 - 2x^3 + x - 8\) is divided by \(x+2\) is 134.