Problem 8 - Algebra 2

Question

If \(p(x)=2 x^{3}+6 x-12\) and \(q(x)=5 x^{2}+4,\) find each value. \(3 p(a)-q(a+1)\)

Step-by-Step Solution

Answer

The answer is \(6a^3 - 5a^2 + 8a - 45\).

Step 1: Understand the Problem

We are given two polynomial functions: \( p(x) = 2x^3 + 6x - 12 \) and \( q(x) = 5x^2 + 4 \). We need to find the expression \( 3p(a) - q(a+1) \). This requires us to first calculate \( p(a) \) and then \( q(a+1) \), and then use these values to compute the expression.

Step 2: Substitute for p(a)

Plug \( x = a \) into \( p(x) \) to find \( p(a) \). This gives us:\[p(a) = 2a^3 + 6a - 12\]

Step 3: Multiply p(a) by 3

Now multiply the entire expression for \( p(a) \) by 3:\[3p(a) = 3(2a^3 + 6a - 12) = 6a^3 + 18a - 36\]

Step 4: Substitute for q(a+1)

Next, substitute \( x = a+1 \) into \( q(x) \) to find \( q(a+1) \). This results in:\[q(a+1) = 5(a+1)^2 + 4\]Expanding \((a+1)^2\):\[q(a+1) = 5(a^2 + 2a + 1) + 4 = 5a^2 + 10a + 5 + 4 = 5a^2 + 10a + 9\]

Step 5: Calculate 3p(a) - q(a+1)

Now, subtract the expression for \( q(a+1) \) from \( 3p(a) \):\[3p(a) - q(a+1) = (6a^3 + 18a - 36) - (5a^2 + 10a + 9)\]Simplify the right-hand side:\[= 6a^3 - 5a^2 + 18a - 10a - 36 - 9 = 6a^3 - 5a^2 + 8a - 45\]

Step 6: Finalize the Answer

The expression for \( 3p(a) - q(a+1) \) is simplified to \( 6a^3 - 5a^2 + 8a - 45 \).

Key Concepts

Polynomial Expression

A polynomial expression, in simple terms, is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Polynomials can take various forms, including linear, quadratic, cubic, and higher degrees, represented by equations like:

• Constant polynomial - for example, the number 5 or -3.
• Linear polynomial - such as 2x + 3.
• Quadratic polynomial - in the form of x² + 4x + 4.
• Cubic polynomial - like 2x³ + 6x - 12 as seen in the given problem.

Polynomial expressions can include different terms, where each term is composed of a coefficient (a numerical factor) and a variable raised to an exponent. These expressions are algebraically manipulated to solve various mathematical and practical problems. Understanding how to handle polynomials is crucial for evaluating expressions, solving equations, and much more.

Substitution

Substitution refers to replacing a variable in a mathematical expression with a number or another expression to evaluate it or to form another expression. In the step-by-step solution, substitution is used repeatedly:- To find the value of the function at a specific point, we substitute a variable with a number, like replacing x with a in the polynomial expression, turning it into a concrete value: \[ p(a) = 2a^3 + 6a - 12 \]- Similarly, substitution is used to evaluate another polynomial at a different point by plugging in a shifted variable, such as x = a+1 in q(x).Therefore, substitution is a fundamental technique in algebra that aids in converting abstract polynomial expressions into numeric equations or others, facilitating easier computation and manipulation.

Simplification

Simplification is the process of reducing a mathematical expression to its simplest form. This often means combining like terms, performing arithmetic operations, and factoring where applicable. In our example:- When computing \(3p(a)\), we must first find \(2a^3 + 6a - 12\) and then multiply the entire expression by 3: \[ 3p(a) = 6a^3 + 18a - 36 \] This operation reduced the expression to a more straightforward polynomial with clear terms ready for further combination or calculation.- The simplification process in the problem allows us to efficiently calculate the expression \(3p(a) - q(a+1)\), neatly organizing and reducing it to \(6a^3 - 5a^2 + 8a - 45\).Simplification makes the expression more manageable, reducing the complexity and chance of error in subsequent calculations.

Algebraic Manipulation

Algebraic manipulation involves rearranging and altering algebraic expressions to simplify or solve them. It combines various algebraic skills, such as substitution and simplification, to solve equations or express problems differently.In our step-by-step solution:1. **Combining Like Terms:** After substituting and computing individually for \(3p(a)\) and \(q(a+1)\), we subtract and gather similar linear, quadratic, and cubic terms in the expression: \[ 3p(a) - q(a+1) = (6a^3 + 18a - 36) - (5a^2 + 10a + 9) \] becomes organized into a reduced, coherent form: \[ 6a^3 - 5a^2 + 8a - 45 \]2. **Rearranging Expressions:** This involves careful ordering to make the terms easier to handle.Through these techniques, algebraic manipulation streamlines the algebraic process by positioning the expression in its cleanest and most readable form, ultimately leading to an accurate and complete solution.