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Chapter 11: Problem 54

We have seen the close connection between polynomial division and writing a quotient of polynomials in lowest terms after factoring the numerator. We can also show a connection between dividing one polynomial by another and factoring the first polynomial. letting $$ f(x)=2 x^{2}+5 x-12 $$ Evaluate \(f\left(\frac{3}{2}\right)\).

Short Answer

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Step by step solution

01

Substitute the value

Given the function \(f(x) = 2x^2 + 5x - 12\), substitute \(x = \frac{3}{2}\) into the equation.
02

Compute the squared term

Calculate \(\left(\frac{3}{2}\right)^2\) which equals \(\frac{9}{4}\). Then, multiply this by 2: \(2 \cdot \frac{9}{4} = \frac{18}{4} = \frac{9}{2}\).
03

Compute the linear term

Calculate \(5 \cdot \frac{3}{2} = \frac{15}{2}\).
04

Combine and simplify

Now collect all the parts together: \(f\left(\frac{3}{2}\right) = \frac{9}{2} + \frac{15}{2} - 12\).
05

Perform the addition

Add the fractions: \(\frac{9}{2} + \frac{15}{2} = \frac{24}{2} = 12\).
06

Subtract the constant term

Subtract 12 from the previous result: \(12 - 12 = 0\).

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Key Concepts

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

Evaluating Polynomial Functions
Evaluating polynomial functions involves substituting a given value for the variable and simplifying the expression. In the exercise example, we had the function: \( f(x)=2x^{2}+5x-12 \). To evaluate \( f\left(\frac{3}{2}\right) \), we substituted \( x \) with \( \frac{3}{2} \).

Here's the step-by-step process:
  • Substitute \( x = \frac{3}{2} \) into the polynomial function.
    Given the function is \( 2{\left(\frac{3}{2}\right)}^2 + 5\left(\frac{3}{2}\right) - 12 \).
  • Simplify each term separately.
    First, calculate the squared term: \( \left(\frac{3}{2}\right)^2 = \frac{9}{4} \). Then, multiply by 2: \( 2 \cdot \frac{9}{4} = \frac{18}{4} = \frac{9}{2} \).
  • Next, compute the linear term: \( 5 \cdot \frac{3}{2} = \frac{15}{2} \).
  • Combine and simplify all parts: \( f\left(\frac{3}{2}\right) = \frac{9}{2} + \frac{15}{2} - 12 \).
  • Perform the additional step: \( \frac{9}{2} + \frac{15}{2} = \frac{24}{2} = 12 \).
  • Finally, subtract the constant term \( 12 - 12 = 0 \).
This process of evaluating a polynomial helps check specific values and understand the polynomial's behavior for different input values.
Factoring Polynomials
Factoring polynomials is the process of breaking down a polynomial into simpler polynomials that, when multiplied together, give the original polynomial. It is a critical skill in algebra. For example, consider the polynomial \( 2x^2 + 5x - 12 \).

Steps to factor a polynomial:
  • Identify the polynomial to factor. Here, it is \( 2x^2 + 5x - 12 \).
  • Find two numbers that multiply to give the product of the coefficient of \( x^2 \) term (2) and the constant term (-12), and add up to the coefficient of the \( x \) term (5).
    The numbers are 8 and -3 because \( 2 \cdot (-12) = -24 \) and \( 8 + (-3) = 5 \).
  • Rewrite the middle term (5x) using the numbers found: \( 2x^2 + 8x - 3x - 12 \).
  • Group terms to factor by grouping: \( 2x(x + 4) - 3(x + 4) \).
  • Factor out the common factor from each group: \((2x - 3)(x + 4) \).
Factoring helps solve polynomial equations and simplifies expressions, making complex problems more manageable.
Substitution in Algebra
Substitution in algebra involves replacing a variable with a specific value or another expression to evaluate or simplify an algebraic expression. It is widely used in evaluating polynomial functions and solving equations.

In the given exercise, we used substitution to find \( f\left(\frac{3}{2}\right) \) for the polynomial function \( f(x)=2x^{2}+5x-12 \).

Steps to perform substitution:
  • Identify the expression or equation where substitution is needed. Here, it is \( f(x)=2x^{2}+5x-12 \).
  • Replace the variable \( x \) with the given value \( \frac{3}{2} \).
  • Simplify the expression step by step by performing arithmetic operations.
    Compute each term separately, combine them, and simplify if needed.
Substitution is a fundamental technique to evaluate expressions, solve algebraic equations, and verify factorization.
Simplifying Expressions
Simplifying expressions in algebra means combining and reducing an expression to its simplest form. This ensures that it is easy to understand and work with. Simplification often involves combining like terms, factoring, and performing arithmetic operations.

In the given exercise, after substituting \( \frac{3}{2} \) into \( f(x)=2x^{2}+5x-12 \), we simplified the resulting expression step by step.

Steps for simplification:
  • Combine all like terms. In \(\frac{9}{2} + \frac{15}{2} - 12 \), combine \(\frac{9}{2} \) and \(\frac{15}{2} \) as they are like terms (both involving \(\frac{1}{2} \)).
  • Convert to a common denominator if needed to combine fractions.
    Then, \( \frac{9}{2} + \frac{15}{2} = \frac{24}{2} \).
  • Simplify fractions and perform arithmetic operations like addition or subtraction.
    Finally, \( 12 - 12 = 0 \).
Simplifying expressions allows understanding the algebraic structures better, making it easier to solve equations and perform further analysis.

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Most popular questions from this chapter

Graph each rational function. $$f(x)=\frac{1}{(x+5)(x-2)}$$

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