Ch. 3 - Polynomial and Rational Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 3a

Chapter 4, Problem 3a

Divide using long division. State the quotient, and the remainder, r(x). (x³+5x²+7x+2)÷(x+2)

Verified step by step guidance

Step 1: Set up the long division by writing the dividend $x^3 + 5x^2 + 7x + 2$ under the division bar and the divisor $x + 2$ outside the division bar.

Step 2: Divide the leading term of the dividend $x^3$ by the leading term of the divisor $x$. This gives the first term of the quotient, $x^2$. Write $x^2$ above the division bar.

Step 3: Multiply $x^2$ by the divisor $x + 2$, which results in $x^3 + 2x^2$. Subtract $x^3 + 2x^2$ from the dividend $x^3 + 5x^2 + 7x + 2$, leaving $3x^2 + 7x + 2$.

Step 4: Repeat the process with the new dividend $3x^2 + 7x + 2$. Divide the leading term $3x^2$ by $x$, which gives $3x$. Write $3x$ in the quotient. Multiply $3x$ by $x + 2$, resulting in $3x^2 + 6x$. Subtract $3x^2 + 6x$ from $3x^2 + 7x + 2$, leaving $x + 2$.

Step 5: Divide the leading term $x$ by $x$, which gives $1$. Write $1$ in the quotient. Multiply $1$ by $x + 2$, resulting in $x + 2$. Subtract $x + 2$ from $x + 2$, leaving a remainder of $0$. The quotient is $x^2 + 3x + 1$ and the remainder is $0$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polynomial Long Division

Polynomial long division is a method used to divide polynomials, similar to numerical long division. It involves dividing the leading term of the dividend by the leading term of the divisor, multiplying the entire divisor by this result, and subtracting it from the dividend. This process is repeated with the new polynomial until the degree of the remainder is less than the degree of the divisor.

Quotient and Remainder

In polynomial division, the quotient is the result of the division, representing how many times the divisor fits into the dividend. The remainder is what is left over after the division process, which cannot be divided by the divisor anymore. The relationship can be expressed as: Dividend = Divisor × Quotient + Remainder.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression. It determines the polynomial's behavior and the number of roots it can have. Understanding the degree is crucial in polynomial division, as it helps identify when to stop the division process, specifically when the degree of the remainder is less than that of the divisor.

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Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. (x+1)(x−7)≤0

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In Exercises 1–4, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola's axis of symmetry. Use the graph to determine the function's domain and range. $f (x) = - x^{2} + 2 x + 3$

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Use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=3x⁴−11x³−x²+19x+6

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In Exercises 5–6, use the function's equation, and not its graph, to find (a) the minimum or maximum value and where it occurs. (b) the function's domain and its range. $f (x) = - x^{2} + 14 x - 106$

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Textbook Question

Use the four-step procedure for solving variation problems given on page 447 to solve Exercises 1–10. y varies inversely as x. y = 12 when x = 5. Find y when x = 2.

Textbook Question

In Exercises 1–10, determine which functions are polynomial functions. For those that are, identify the degree. g(x)=6x⁷+πx⁵+2/3 x

Divide using long division. State the quotient, and the remainder, r(x). (x3+5x2+7x+2)÷(x+2)

Verified video answer for a similar problem:

Key Concepts

Polynomial Long Division

Quotient and Remainder

Degree of a Polynomial

Divide using long division. State the quotient, and the remainder, r(x). (x³+5x²+7x+2)÷(x+2)