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Magazine Chapter 3: Problem 42
Use synthetic division to find \(P(k)\) $$k=\sqrt{3} ; \quad P(x)=x^{4}+2 x^{2}-10$$
Short Answer
Expert verified
The remainder is 5, so \( P(\sqrt{3}) = 5 \).
Step by step solution
01
Set Up Synthetic Division
First, set up the synthetic division table with the root \( k = \sqrt{3} \) and the coefficients of \( P(x) = x^4 + 0x^3 + 2x^2 + 0x - 10 \). The coefficients are: \( 1, 0, 2, 0, -10 \). Position \( \sqrt{3} \) to the left of the table.
02
Bring Down the Leading Coefficient
Bring down the leading coefficient, which is 1, to the bottom row.
03
Multiply and Add
Multiply \( \sqrt{3} \) by the value just brought down (1) and write the result (\( \sqrt{3} \)) in the next column of top row. Add this to the coefficient in the next column (0), yielding \( \sqrt{3} \), which goes to the bottom row.
04
Repeat Multiplication and Addition
Multiply \( \sqrt{3} \) by \( \sqrt{3} \) (bottom row value) which equals 3. Add this to the next column's coefficient (2) to get 5, which is placed in the bottom row.
05
Continue Process
Continue the process: multiply 5 by \( \sqrt{3} \) to get \( 5\sqrt{3} \). Add to the next column's coefficient (0) to get \( 5\sqrt{3} \). Write this in the bottom row.
06
Final Multiplication and Addition
Multiply \( 5\sqrt{3} \) by \( \sqrt{3} \) to get 15. Add to the final coefficient (\(-10\)) to get 5. This is the remainder.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polynomial Evaluation
Polynomial evaluation is a process that helps us find the value of a polynomial for a specific variable input. In simpler terms, it is plugging in numbers for the variable and calculating the result.
To evaluate a polynomial, you can use different methods, such as direct substitution, synthetic division, or the remainder theorem. Synthetic division is especially useful when the polynomial is divided by a linear expression.
To evaluate a polynomial, you can use different methods, such as direct substitution, synthetic division, or the remainder theorem. Synthetic division is especially useful when the polynomial is divided by a linear expression.
- Example: To evaluate the polynomial \(P(x) = x^4 + 2x^2 - 10\) at \(x = \sqrt{3}\), you substitute \(\sqrt{3}\) in place of \(x\).
- Direct Substitution: Replace each occurrence of the variable with the given value and simplify the expression.
- Use Synthetic Division: Synthetic division simplifies finding \(P(k)\) by systematically dividing the polynomial.
Roots of Polynomial
When we talk about the roots of a polynomial, we are referring to the values that make the polynomial equal to zero. In other words, these are the solutions to the polynomial equation.
- Finding Roots: Calculating the roots involves solving the equation \(P(x) = 0\). This may require factoring, using the quadratic formula, or other algebraic methods.
- Role in Synthetic Division: Roots are crucial in synthetic division because they serve as the divisor. If a polynomial division yields a remainder of zero when synthetic division is applied with a potential root \(k\), then \(k\) is a confirmed root.
- Significance: Knowing the roots of a polynomial allows us to understand its behavior, such as where it crosses the x-axis and how it factors into simpler terms.
Algebraic Expression
Algebraic expressions are combinations of numbers, variables, and operational symbols (+, 鈭�, 脳, 梅).Polynomials are a specific type of algebraic expression characterized by variables raised to whole number powers with constant coefficients.
- Components of Polynomials: Polynomials consist of terms that can include constants, variables, and coefficients. Each term is a product of a constant and a non-negative integer power of a variable.
- Example: In \(P(x) = x^4 + 2x^2 - 10\), the terms are \(x^4\), \(2x^2\), and \(-10\).
- Simplifying Expressions: Simplifying involves combining like terms and reducing complex expressions. This is often necessary before applying synthetic division or finding roots.
