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Chapter 5: Problem 6

Evaluate each polynomial for \(x=2\) and \(y=-3\). $$3 x^{2} y-4 y+5$$

Short Answer

Expert verified
The value of the polynomial \(3x^{2}y-4y+5\) for \(x=2\) and \(y=-3\) is -19.

Step by step solution

01

Substitution

First, replace the variables \(x\) and \(y\) in the polynomial \(3x^{2}y-4y+5\) with their corresponding values \(x=2\) and \(y=-3\). This gives us the equation \(3(2)^{2}(-3)-4(-3)+5\).
02

Simplification Using BODMAS

Keeping in mind BODMAS, start with the exponentiation, resulting in \(3(4)(-3)-4(-3)+5\). The product operation follows, giving us \(-36+12+5\). Finally, perform the addition and subtraction operations to get the value of the polynomial: \(-36+12+5 = -19\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution Method
The substitution method is a key concept in algebra, especially when it comes to evaluating polynomials. To evaluate a polynomial such as \(3x^{2}y-4y+5\) for specific values of variables \(x\) and \(y\), you begin by substituting these values into the polynomial. Here, it's given that \(x = 2\) and \(y = -3\).
By substituting, the polynomial transforms into \(3(2)^{2}(-3) - 4(-3) + 5\). This simple step is powerful because it takes the abstract mathematical expression and turns it into a numerical equation that can be solved.
  • Identify the values to substitute: \(x = 2\) and \(y = -3\)
  • Replace each instance of \(x\) and \(y\) in the polynomial with these values
  • The equation becomes numeric, paving the way for straightforward calculations
This method underlines the importance of replacing variables with given numbers, resulting in a simpler problem that can be easily computed.
BODMAS Rule
When working through polynomial expressions, the BODMAS rule ensures you perform operations in the correct order. BODMAS stands for Brackets, Orders (i.e., powers and roots, etc.), Division and Multiplication (from left to right), Addition and Subtraction (from left to right). This rule is crucial when dealing with complex equations involving multiple operations.
For our given polynomial, after substitution, it reads \(3(2)^{2}(-3) - 4(-3) + 5\). Applying BODMAS:
  • Address any brackets, though here they simply indicate multiplication
  • Compute the order by solving \((2)^{2}\), which equals 4
  • Proceed with multiplication: first, \(3 \times 4 \times (-3)\) results in \(-36\)
  • Next, \(-4 \times (-3)\) yields 12
  • Finally, complete the addition and subtraction from left to right: \(-36 + 12 + 5\)
Observing the BODMAS rule ensures the operations are performed logically and that the most efficient path to solving the polynomial is followed, ultimately arriving at the correct result.
Simplification Process
The simplification process is the final stage when evaluating polynomial expressions. After correctly substituting and applying BODMAS, simplifying is about reducing the expression to its simplest form. It can involve combining like terms and performing basic arithmetic.
In the expression \(-36 + 12 + 5\), simplification entails:
  • First, adding the positive numbers \(12 + 5\) to obtain \(17\)
  • Then subtracting \(36\) from \(17\) to find the final solution, \(-19\)
Simplification is about making the expression more comprehensible and resolving it to a single, straightforward value. This not only finalizes the solution but also verifies the accuracy of the operations carried out in previous steps.

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Most popular questions from this chapter

When dividing a binomial into a polynomial with missing terms, explain the advantage of writing the missing terms with zero coefficients.

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