Problem 98 - Intermediate Algebra

Question

$$ \left(5 x^{4}-3 x^{2}+2\right) \div(x+2) $$

Step-by-Step Solution

Answer

Quotient: \(5x^3 - 10x^2 + 17x - 34\), Remainder: \(70\)

Step 1: Set Up Polynomial Long Division

To perform polynomial long division, we first set up the division by writing the dividend \(5x^4 - 3x^2 + 2\) under a division bracket and \(x + 2\) outside the bracket. Make sure to include any terms with zero coefficients if needed, like \(0x^3\).

Step 2: Divide the Leading Terms

Divide the leading term of the dividend \(5x^4\) by the leading term of the divisor \(x\). This gives \(5x^3\). Write \(5x^3\) above the division bracket as the first term of the quotient.

Step 3: Multiply and Subtract

Multiply \(5x^3\) by the entire divisor \(x + 2\) to get \(5x^4 + 10x^3\). Subtract \(5x^4 + 10x^3\) from the current dividend to obtain \(-10x^3 - 3x^2 + 2\). Bring down the next term if needed.

Step 4: Repeat Division Steps

Divide the new leading term \(-10x^3\) by \(x\) to get \(-10x^2\). Multiply \(-10x^2\) by \(x + 2\) to get \(-10x^3 - 20x^2\), and subtract this from the current dividend, resulting in \(17x^2 + 2\).

Step 5: Continue Dividing

Divide \(17x^2\) by \(x\), which gives \(17x\). Multiply \(17x\) by \(x + 2\) to get \(17x^2 + 34x\). Subtract to get the new dividend \(-34x + 2\).

Step 6: Final Division Steps

Divide \(-34x\) by \(x\) to obtain \(-34\). Multiply \(-34\) by \(x + 2\) resulting in \(-34x - 68\). Subtract to find the remainder, which is \(70\).

Step 7: Write the Quotient and Remainder

The quotient from the division is \(5x^3 - 10x^2 + 17x - 34\) and the remainder is \(70\). Therefore, the result of the division is \(5x^3 - 10x^2 + 17x - 34 + \frac{70}{x+2}\).

Key Concepts

Polynomial Long Division

Polynomial long division is a method similar to long division with numbers, but it applies to polynomials. It is used to divide a polynomial (dividend) by another polynomial (divisor), which are generally in terms of variables. To set up, write the polynomial you are dividing (the dividend) under the division bracket and the polynomial you are dividing by (the divisor) is placed outside of it.

• Always arrange the terms of each polynomial in descending order of powers for clarity.
• Include placeholder terms with zero coefficients if any power of the variable is missing in the dividend.

This setup allows you to focus on dividing each part of the dividend by the divisor step-by-step until all parts have been divided.

Quotient and Remainder Theorem

The Quotient and Remainder Theorem is pivotal in polynomial division. It states that for any polynomials, the dividend can be represented as the product of the divisor and the quotient, plus the remainder. Mathematically, if dividing a polynomial \( P(x) \) by \( D(x) \), you will get:\[P(x) = D(x) \times Q(x) + R(x)\]where \( Q(x) \) is the quotient and \( R(x) \) (remainder) is of a lower degree than \( D(x) \).

• The remainder's degree is always less than the degree of the divisor.
• The remainder can be zero, in which case the division is exact.

In our example, the quotient is \(5x^3 - 10x^2 + 17x - 34\) and the remainder is \(70\).

Leading Terms

Leading terms play a crucial role when dividing polynomials. The leading term of a polynomial is the term with the highest degree, and it guides the division process. Begin each step in polynomial long division by dividing the leading term of the current dividend by the leading term of the divisor. This process helps determine each term in the quotient.

• In our example, starting with \(5x^4\) (the leading term of the dividend) divided by \(x\) (the leading term of the divisor), gives the first term of the quotient, \(5x^3\).
• This approach is repeated for each new polynomial expression formed after subtraction, effectively reducing the polynomial until all terms are processed.

Understanding leading terms helps in consistently applying division in a structured manner.

Dividing Polynomials

Dividing polynomials involves repeatedly applying the division of leading terms and subtracting the resulting polynomial expressions. Each division reduces the polynomial's degree until you find a remainder smaller than the divisor.

• Start by dividing the leading term of the dividend by the leading term of the divisor to find the first term of the quotient.
• Multiply this term back by the entire divisor and subtract from the dividend.
• Bring down any terms left from the original dividend or move to the next term if necessary, and repeat the process with the new leading term.

This repetitive cycle continues until the entire polynomial is divided as much as possible. The end result gives a quotient and possibly a remainder.