Problem 93 - Algebra A Combined Function

Question

Fill in the squares so that a true statement forms. $$ \left(x^{\square}+7\right)\left(x^{\square}+3\right)=x^{4}+10 x^{2}+21 $$

Step-by-Step Solution

Answer

Both squares should be filled with 2.

Step 1: Analyze the given polynomial

We are given a polynomial multiplication: \( (x^{\square}+7)(x^{\square}+3) = x^4 + 10x^2 + 21 \). The goal is to fill in the squares so the expression forms a correct equation.

Step 2: Expand the general form

Let's denote the squares as \(x^a\) and \(x^b\). Expanding the expression, we have: \[ (x^a + 7)(x^b + 3) = x^{a+b} + 3x^a + 7x^b + 21. \] Simplifying, the expression becomes \(x^{a+b} + (3x^a + 7x^b) + 21 \).

Step 3: Match the expanded polynomial with the target

The expanded form \(x^{a+b} + (3x^a + 7x^b) + 21\) must match \(x^4 + 10x^2 + 21\). Two relationships arise: \(a+b = 4\) and \(3x^a + 7x^b = 10x^2\).

Step 4: Solve for the exponents

From \(a+b=4\) and \(3x^a + 7x^b = 10x^2\), we deduce both \(a = 2\) and \(b = 2\) since that satisfies \(3x^2 + 7x^2 = 10x^2\), ensuring all terms balance.

Key Concepts

Polynomial Expansion

Polynomial expansion involves multiplying expressions to simplify or express them in an extended form. For example, when dealing with polynomials, each term in the first polynomial is multiplied by each term in the second polynomial.

In the given problem, you need to understand how expansion works in a practical context. Consider the expression \((x^a + 7)(x^b + 3)\). To expand this, multiply every term in the first bracket by every term in the second bracket:

• First, multiply \(x^a\) by \(x^b\) to get \(x^{a+b}\).
• Then \(x^a\) comes next with the constant from the second bracket, turning into \(3x^a\).
• Next, tackle the 7 from the first bracket, giving you \(7x^b\) when multiplied by \(x^b\).
• Finally, multiply the constants 7 and 3 to yield 21.

Combining these results, you arrange the expanded form: \[ x^{a+b} + 3x^a + 7x^b + 21 \].

This detailed step-by-step multiplication is crucial as it shows precisely which parts of the polynomial contribute to different terms in the final expression.

Exponent Rules

Exponents are an essential part of algebra, particularly in expressions where terms are raised to specific powers. Understanding exponent rules helps solve polynomial equations effectively.

Key rules include:

• Product of Powers Rule: When multiplying terms with the same base, add the exponents. For instance, \(x^a \times x^b = x^{a+b}\). This is critical for grouping terms that result in higher powers like we see in the polynomial expansion.
• Power of a Power Rule: If a term with an exponent is raised by another power, multiply the exponents. Thus, \((x^a)^b = x^{ab}\).
• Zero Exponent Rule: Any non-zero number raised to the power of 0 equals 1, making \(x^0 = 1\) a fundamental concept in simplifying expressions.

In the exercise, recognizing that \(a + b = 4\) makes it possible to solve for these exponent values, marrying both the power rule and your understanding of how terms combine to match the target polynomial.

Algebraic Expressions

Algebraic expressions consist of variables, constants, and coefficients combined using operations such as addition, subtraction, and multiplication. Unlike equations, expressions do not include equality between two sides.

To master algebraic expressions within the realm of polynomials:

• Understand the Components: Every polynomial includes terms, such as \(x^a\), where \(x\) is a variable and \(a\) is the exponent.
• Learn Coefficient Matching: In expanding literal polynomial expressions, aligning coefficients after expansion helps ensure the equality to a given polynomial.
• Simplification: Combine like terms to reduce expressions to their simplest form, aiding in matching aspects such as in the sample solution \(3x^2 + 7x^2\) becomes \(10x^2\).

These skills aid in understanding how algebra formulates expressions that represent more complex relationships, useful in identifying the value of variables or unknowns.