Problem 50 - Algebra 2

Question

If \(p(x)=2 x^{2}-5 x+4\) and \(r(x)=3 x^{3}-x^{2}-2,\) find each value. $$ p\left(x^{2}+4\right) $$

Step-by-Step Solution

Answer

\(p(x^2+4) = 2x^4 + 11x^2 + 16\)

Step 1: Understand the expression

We need to find the value of the polynomial \( p(x) = 2x^2 - 5x + 4 \) when \( x \) is replaced by \( x^2 + 4 \). This means we substitute \( x^2 + 4 \) in place of \( x \) in the polynomial.

Step 2: Substitute \( x \) with \( x^2 + 4 \) in \( p(x) \)

Replace every \( x \) in the polynomial \( p(x) = 2x^2 - 5x + 4 \) with \( (x^2 + 4) \). So, it becomes:\[p(x^2+4) = 2(x^2+4)^2 - 5(x^2+4) + 4\]

Step 3: Expand \((x^2 + 4)^2\)

Calculate \((x^2 + 4)^2\):\[(x^2 + 4)^2 = x^4 + 8x^2 + 16\]

Step 4: Substitute the expanded form into the expression

Use the expanded polynomial to continue the substitution:\[p(x^2+4) = 2(x^4 + 8x^2 + 16) - 5(x^2 + 4) + 4\]

Step 5: Distribute the constants

Distribute 2 and -5 in their respective expressions:\[p(x^2+4) = 2x^4 + 16x^2 + 32 - 5x^2 - 20 + 4\]

Step 6: Combine like terms

Now add or subtract like terms together:\[p(x^2+4) = 2x^4 + (16x^2 - 5x^2) + (32 - 20 + 4)\]This simplifies to:\[p(x^2+4) = 2x^4 + 11x^2 + 16\]

Key Concepts

Polynomial Substitution

Polynomial substitution involves replacing the variable in a polynomial with another algebraic expression. This technique is crucial when solving problems that require the evaluation of polynomials at different points or expressions. In our example, we have the polynomial \( p(x) = 2x^2 - 5x + 4 \). The exercise requires substituting \( x \) with \( x^2 + 4 \).

To substitute, follow these steps:

• Identify the parts of the polynomial where the substitution will occur, which in this case are the occurrences of \( x \) in \( p(x) \).
• Replace every \( x \) in the polynomial with \( x^2+4 \), resulting in \( p(x^2+4) = 2(x^2+4)^2 - 5(x^2+4) + 4 \).

This method translates the problem into a new expression that we can further simplify.

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operations. In our context, \( x^2 + 4 \) and the polynomial \( p(x) = 2x^2 - 5x + 4 \) are examples of algebraic expressions. Understanding how to work with such expressions is fundamental to algebra.

• Variables in expressions can be replaced or substituted as needed, which is a key part of solving polynomial expressions.
• The operation of substitution doesn't change the original expression's value if the substitute is handled correctly.

Algebraic expressions serve as the building blocks for constructing complex functions and descriptions of numerical relationships. Knowing how to manipulate them by substitution is essential in more advanced mathematical contexts.

Polynomial Expansion

Polynomial expansion involves taking a binomial or a polynomial raised to a power and expanding it into a longer algebraic expression. In this problem, we expand \( (x^2 + 4)^2 \) to move forward with the evaluation.

• When expanding \( (x^2 + 4)^2 \), we apply the distributive property, multiplying each term in one bracket by each term in the other, giving \( x^4 + 8x^2 + 16 \).
• This expansion is critical because it simplifies the substitution, allowing for further calculations and reductions.

Understanding polynomial expansion helps manage expressions as they grow in complexity and ensures accurate computations.

Combining Like Terms

Combining like terms refers to the process of simplifying algebraic expressions by consolidating terms that have the same variables raised to the same power. This step is crucial in our solution to reach a final simplified expression.

• After performing all substitutions and expansions, we look at \( 2x^4 + 16x^2 + 32 - 5x^2 - 20 + 4 \). This expression includes like terms, such as the \( x^2 \) terms \( 16x^2 \) and \( -5x^2 \).
• Adding or subtracting these terms results in simplified forms: \( 16x^2 - 5x^2 = 11x^2 \) and \( 32 - 20 + 4 = 16 \).
• Finally, combining yields the result \( 2x^4 + 11x^2 + 16 \).

Mastering the combination of like terms is essential for reducing expressions, making them manageable and interpretable.