Problem 70 - Algebra and Trigonometry

Question

Nested Form of a Polynomial 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

Polynomials P(x) and Q(x) are the same. R(3) evaluated in nested form is 157.

Step 1: Apply Distributive Property

Begin with the given nested polynomial expression for \( Q(x) = (((3x-5)x+1)x-3)x+5 \). Start by expanding the innermost expression using the distributive property. First, expand \((3x-5)x\): \( (3x-5)x = 3x^2 - 5x \).

Step 2: Expand Further

Take the result from Step 1, \(3x^2 - 5x\), and substitute it back into the next part of the nested expression: \(((3x^2 - 5x) + 1)\). Multiply this by \(x\): \((3x^2 - 5x + 1)x = 3x^3 - 5x^2 + x\).

Step 3: Continue Expanding

Add \(-3\) to the expression from Step 2: \((3x^3 - 5x^2 + x - 3)\). Multiply the entire expression by \(x\): \((3x^3 - 5x^2 + x - 3)x = 3x^4 - 5x^3 + x^2 - 3x\).

Step 4: Final Expansion Step

Lastly, add 5 to the result of Step 3: \(3x^4 - 5x^3 + x^2 - 3x + 5\). This is the expanded expression for \(Q(x)\).

Step 5: Comparing P(x) and Q(x)

The expanded form of \(Q(x)\) is the same as \(P(x) = 3x^4 - 5x^3 + x^2 - 3x + 5\). Thus, \(P(x)\) and \(Q(x)\) are the same polynomial.

Step 6: Evaluate P(2) and Q(2)

To evaluate \(P(2)\) and \(Q(2)\), substitute \(x = 2\): \(P(2) = 3(2)^4 - 5(2)^3 + 2^2 - 3(2) + 5 = 48 - 40 + 4 - 6 + 5 = 11\). For \(Q(2)\), follow the nested evaluation: \(Q(2) = (((3(2) - 5)2 + 1)2 - 3)2 + 5 = (((6 - 5)2 + 1)2 - 3)2 + 5 = (((1)2 + 1)2 - 3)2 + 5 = ((2 + 1)2 - 3)2 + 5 = (3*2 - 3)2 + 5 = (6 - 3)2 + 5 = 3*2 + 5 = 6 + 5 = 11\). Both evaluations yield 11, confirming the equivalence.

Step 7: Transform R(x) into Nested Form

To write \(R(x) = x^5 - 2x^4 + 3x^3 - 2x^2 + 3x + 4\) in nested form, apply the nested method. Factor incrementally: \(R(x) = (((((x - 2)x + 3)x - 2)x + 3)x + 4)\).

Step 8: Evaluate R(3) Using Nested Form

Evaluate using nested form: \(R(3) = (((((3 - 2)3 + 3)3 - 2)3 + 3)3 + 4) = ((((1·3 + 3)3 - 2)3 + 3)3 + 4) = (((3 + 3)3 - 2)3 + 3)3 + 4 = ((6·3 - 2)3 + 3)3 + 4 = (18 - 2)3 + 3)3 + 4 = (16·3 + 3)3 + 4 = (48 + 3)3 + 4 = 51·3 + 4 = 153 + 4 = 157\). This demonstrates ease and efficiency, correlating with synthetic division steps.

Key Concepts

Nested Form

The nested form of a polynomial is an efficient way to express otherwise lengthy polynomial equations. It involves expressing the polynomial in a way that continues to build on the previous expression. In the expression for the polynomial \( Q(x) = (((3x-5)x+1)x-3)x+5 \), you see how each term is built from the inside out. This structure makes it easier to evaluate polynomials mentally, as fewer computations are interrupted by constant terms. To convert a polynomial into nested form, start with the highest degree terms and progressively factor out \( x \) values beginning from the inside. For example, \( R(x) = x^5 - 2x^4 + 3x^3 - 2x^2 + 3x + 4 \) can be expressed in nested form as \( (((((x - 2)x + 3)x - 2)x + 3)x + 4) \). Each component is transformed by focusing first on reducing it to a simpler set of operations before moving outward.The nested form not only showcases the polynomial distinctly but also streamlines the evaluation process, occurring seamlessly in your mind with practice.

Distributive Property

The distributive property is a fundamental concept in algebra that helps in expanding and simplifying expressions. It refers to the rule that states you can distribute multiplication over addition or subtraction. Mathematically, it can be represented as \( a(b + c) = ab + ac \).In the context of polynomials, applying the distributive property means multiplying terms or combining expressions efficiently. Let's consider the expansion of \( Q(x) = (((3x-5)x+1)x-3)x+5 \) in the exercise. We use the distributive property to expand \( (3x-5)x \) to \( 3x^2 - 5x \). This method is iteratively applied, revealing the simplicity behind Nestled expressions.The advantage of this property in polynomial expression is significant. It simplifies complex nested expressions and helps in visually breaking down a polynomial statement into manageable algebraic parts. This allows smoother calculations and helps recognize polynomial identities.

Synthetic Division

Synthetic division is an algebraic technique used for dividing a polynomial by a binomial of the form \( (x - c) \). This method provides a streamlined approach compared to traditional long division and is particularly useful for polynomials.To illustrate, let's calculate \( R(3) \) using its nested form: \( (((((3 - 2)3 + 3)3 - 2)3 + 3)3 + 4) \). Each step resembles the steps in synthetic division, where values are synthesized and computed in succession without expanding polynomial expressions traditionally. Synthetic division eliminates repetitive steps and provides direct insight into polynomial behavior during division. It focuses on coefficients and uses recursive computation, aligning with the nested form's arithmetic steps. By recognizing these parallels, synthetic division becomes a mental exercise similar in practicality to the nested form when evaluating polynomials efficiently.