Problem 22 - A Graphical Approach to College Algebra

Question

Find each quotient when \(P(x)\) is divided by the binomial following it. $$P(x)=x^{4}+4 x^{3}+2 x^{2}+9 x+4 ; \quad x+4$$

Step-by-Step Solution

Answer

The quotient is \(x^3 + 2x + 1\).

Step 1: Set up the Division

To find the quotient of the polynomial \(P(x) = x^4 + 4x^3 + 2x^2 + 9x + 4\) when divided by \(x+4\), we will use polynomial long division. Set up \(x+4\) as the divisor and \(x^4 + 4x^3 + 2x^2 + 9x + 4\) as the dividend.

Step 2: Divide the Leading Terms

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

Step 3: Multiply and Subtract

Multiply \(x^3\) by \(x+4\) to get \(x^4 + 4x^3\). Subtract this from the original polynomial to find the new dividend: \((x^4 + 4x^3 + 2x^2 + 9x + 4) - (x^4 + 4x^3)\) results in \(2x^2 + 9x + 4\).

Step 4: Repeat the Process

Divide the first term of the new dividend, \(2x^2\), by \(x\) to get \(2x\). Multiply \(2x\) by \(x+4\) to get \(2x^2 + 8x\). Subtract \(2x^2 + 8x\) from \(2x^2 + 9x + 4\), leading to \(x + 4\).

Step 5: Continue Division

Divide \(x\) by \(x\) to get \(1\). Multiply \(1\) by \(x+4\), giving \(x + 4\). Subtract \(x + 4\) from \(x + 4\), resulting in \(0\) for the remainder.

Step 6: Write the Final Quotient

The quotient from the division of \(P(x)\) by \(x+4\) has thus been determined to be \(x^3 + 2x + 1\) as the remainder is zero.

Key Concepts

Polynomial Division

Polynomial division is a method used to divide one polynomial by another, much like long division with numbers. It's a systematic procedure allowing us to simplify expressions and determine when a polynomial is divisible by another.

• Setup: Write the dividend (the polynomial you're dividing) and the divisor (the polynomial you're dividing by).
• Dividing: Start with the leading terms. Divide the first term of the dividend by the first term of the divisor.
• Multiply and Subtract: Multiply the result by the entire divisor, subtracting it from the dividend to form a new, smaller polynomial. Repeat as needed.

This method helps find the quotient and remainder when dividing polynomials. If the remainder is zero, the dividend is perfectly divisible by the divisor.

Remainder Theorem

The Remainder Theorem provides a quick way to evaluate the remainder of a polynomial division without performing the entire division. If a polynomial \(f(x)\) is divided by a binomial \((x-c)\), the remainder is simply \(f(c)\).

• This means substituting the constant \(c\) into the polynomial.
• If \(f(c)=0\), then \((x-c)\) is a factor of the polynomial.

The theorem offers an efficient shortcut: no need to perform full division when you're interested only in the remainder. It is especially helpful when checking potential factors of polynomials.

Synthetic Division

Synthetic division is a simplified method of dividing a polynomial by a binomial of the form \((x-c)\). It's quicker and less cumbersome than polynomial long division. This technique is effective for polynomials with leading coefficients of 1.

• Setup: List the coefficients of the polynomial in descending order of degree.
• Bring Down the First Coefficient: Write this directly across in the result row.
• Multiply and Add: Multiply the coefficient by \(c\) (from \(x-c\)) and add to the next coefficient. Continue this process through the coefficients.

Synthetic division results in the coefficients of the quotient polynomial, plus a remainder if there is one. It simplifies the division process, making it a popular technique for quick calculations.

Factoring Polynomials

Factoring polynomials involves rewriting them as a product of simpler polynomials, which can reveal the roots of the polynomial or simplify the expression for further calculation.

• Identify Common Factors: Factor out the greatest common factor from all terms.
• Factor by Grouping: Combine terms to form groups that can be factored separately.
• Special Forms: Recognize patterns like difference of squares, perfect square trinomials, and sum/difference of cubes.

Factoring is crucial for solving polynomial equations and simplifying complex expressions. It links the polynomial to its roots, providing insight into its properties and behaviors.