Problem 68 - Algebra and Trigonometry

Question

Expand \(Q\) to prove that the polynomials \(P\) and \(Q\) are the same. $$\begin{array}{l}{P(x)=3 x^{4}-5 x^{3}+x^{2}-3 x+5} \\ {Q(x)=(((3 x-5) x+1) x-3) x+5}\end{array}$$ Try to evaluate \(P(2)\) and \(Q(2)\) in your head, using the forms given. Which is easier? Now write the polynomial \(R(x)=x^{5}-2 x^{4}+3 x^{3}-2 x^{2}+3 x+4\) in "nested" form, like the polynomial \(Q .\) Use the nested form to find \(R(3)\) in your head. Do you see how calculating with the nested form follows the same arithmetic steps as calculating the value of a polynomial using synthetic division?

Step-by-Step Solution

Answer

P(x) and Q(x) are the same polynomial. Evaluating in nested form is easier for mental calculation.

Step 1: Expand Q(x)

We start with the nested form of \(Q(x)\): \[Q(x)=(((3x-5)x+1)x-3)x+5\]Let's expand it step by step:1. \((3x-5)x=3x^2-5x\)2. \((3x^2-5x)+1=3x^2-5x+1\)3. \((3x^2-5x+1)x=3x^3-5x^2+x\)4. \((3x^3-5x^2+x)-3=3x^3-5x^2+x-3\)5. \((3x^3-5x^2+x-3)x=3x^4-5x^3+x^2-3x\)6. \((3x^4-5x^3+x^2-3x)+5=3x^4-5x^3+x^2-3x+5\)After expanding, \(Q(x)\) is the same as \(P(x)\):\[3x^4-5x^3+x^2-3x+5\]Hence, \(P\) and \(Q\) are the same polynomial.

Step 2: Evaluate P(2) and Q(2)

Let's evaluate \(P(2)\): \[3(2)^4 - 5(2)^3 + (2)^2 - 3(2) + 5 = 3(16) - 5(8) + 4 - 6 + 5 = 48 - 40 + 4 - 6 + 5 = 11\]Now evaluate \(Q(2)\) in the nested form: \[((3(2)-5)2+1)2-3)2+5 = ((6-5)2+1)2-3)2+5 = (1\times2+1)2-3)2+5 = (2+1)2-3)2+5 = (3\times2-3)2+5 = (6-3)2+5 = 3\times2+5 = 6+5 = 11\]Thus, both \(P(2)\) and \(Q(2)\) give the same value, and the nested form makes mental calculation simpler.

Step 3: Write R(x) in Nested Form

We'll express \(R(x) = x^5 - 2x^4 + 3x^3 - 2x^2 + 3x + 4\) in nested form:\[R(x) = ((((x - 2)x + 3)x - 2)x + 3)x + 4\]This breaks down the polynomial by factoring out a common \(x\) from each term iteratively.

Step 4: Evaluate R(3) Using Nested Form

Use the nested form of \(R(x)\) to find \(R(3)\):\[((((3 - 2)3 + 3)3 - 2)3 + 3)3 + 4\]1. \(3 - 2 = 1\)2. \(1 \times 3 + 3 = 6\)3. \(6 \times 3 - 2 = 16\)4. \(16 \times 3 + 3 = 51\)5. \(51 \times 3 + 4 = 157\)So, \(R(3) = 157\). The nested form simplifies mental calculation similarly to synthetic division.

Key Concepts

Synthetic Division

Synthetic division is a simplified method for dividing polynomials, particularly when dividing by a linear factor of the form \(x - c\). Instead of using long division, synthetic division allows for quicker calculations and is especially useful when finding roots of polynomials.

• Synthetic division is more straightforward than traditional long division.
• It involves only the coefficients of the polynomial, streamlining the computation process.
• It ultimately provides the quotient and the remainder without the need for extensive writing.

In synthetic division, you write down the coefficients of the polynomial and apply a series of arithmetic steps with the divisor's root (the "\(c\)" in \(x-c\)). Each multiplication and addition operation helps determine the result iteratively, mimicking mental calculation patterns.

Nested Form

Nested form, also known as Horner's method, is a way of expressing a polynomial to simplify evaluation, and it is particularly handy in mental calculation.

• Nesting involves grouping terms into sub-expressions, making the evaluation process more manageable.
• It reduces the number of operations needed because it reorganizes the polynomial.

For example, consider a polynomial \(Q(x)\): \[Q(x)=(((3x-5)x+1)x-3)x+5\]The nested form restructures the polynomial in a way that closely resembles the arithmetic sequence used in synthetic division, but focuses more on step-by-step evaluation rather than division. This technique helps maintain precision and ease in calculations.

Polynomial Evaluation

Polynomial evaluation refers to the process of calculating the value of a polynomial at a given point. Depending on the form of the polynomial, the evaluation can vary in complexity.

• Direct evaluation uses the polynomial's expanded form, involving straightforward substitution of the value.
• Using nested form or synthetic division can simplify calculations by minimizing operations.

When evaluating a polynomial such as \(P(x)\) at \(x = 2\), either direct substitution or restructured methods like nested form can be used. Each approach can offer efficiency and reduce computational errors, with the nested form often showcasing its strength by breaking down into sub-calculations.

Mental Calculation

Mental calculation techniques are essential when working through polynomial evaluations without paper or calculators, especially if structured approaches are used.

• Techniques like using nested form or synthetic division can greatly aid in performing accurate mental calculations.
• With practice, understanding the step-by-step arithmetic used in these methods can lead to quick evaluations.

Evaluating \(Q(2)\) using nested form as shown in the exercise demonstrates the potential of mental calculation methods. These techniques focus on simplification and reliability, minimizing the burden of keeping track of intermediate results. This usefulness becomes even more apparent in teaching scenarios or fast-paced problem-solving.